Algebra and Trigonometry (1st Edition)

Chapter 12

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; The vertices of the ellipse is of the form and... more

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  ; Since the coordinate of vertices and foci are... more

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Center: Vertices: Co-vertices: .Foci: . ; Since , ... more

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Center: Vertices: Co-vertices: Foci: ; Divide the ... more

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From the given equation of the ellipse , first ... more

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The standard form of the ellipse is .Centre is ... more

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An ellipse is the locus of all those points such ... more

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The foci always lie on the major axis of the ... more

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Circle ; In case of a circle, the minor and major ... more

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The foci lie on the center of the circle. ; In the... more

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An ellipse is symmetrical from the major axis at ... more

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Not an ellipse. ; The given equation  cannot be ... more

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Ellipse ; Convert the given equation into a ... more

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Not an ellipse. ; The given equation is an ... more

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Ellipse ; Convert the given equation into the ... more

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Ellipse ; Convert the given equation into the ... more

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End points of major axis: End points of minor axis... more

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End points of the major axis: End points of the ... more

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The endpoints of the major axes will be , and .The... more

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Endpoints of the major axes will be and .... more

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.Endpoints of the major axis are and .Endpoints ... more

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.Endpoints of the major axes are and .Endpoints ... more

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.Endpoints of the major axes are and .Endpoints ... more

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Endpoints of the major axes are and .Endpoints of... more

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The endpoints of the major axes are and .The ... more

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.The endpoints of the major axes are and .The ... more

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The endpoints of the major axes are and .The ... more

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The endpoints of the major axes are and .The ... more

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The endpoints of the major axes are and .The ... more

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.The endpoints of major axes are and .The ... more

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The endpoints of the major axes are and .The ... more

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The endpoints of the major axes are and .The ... more

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  ; The equation of the ellipse already exists in ... more

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, ; The equation of the ellipse exists in the ... more

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; For the ellipse centered at , the length of ... more

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and ; Use the method of completing the square by ... more

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and ; Use the method of completing the square by ... more

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Write the equation of the ellipse such that is ... more

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In the given equation , , which means that the ... more

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The given equation is rearranged to write it in ... more

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The given equation is rearranged to write it in ... more

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In the given equation , , which means that the ... more

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In the given equation , the denominators are the ... more

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In the given equation , , which means the major ... more

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The given equation is converted into the standard ... more

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The given equation is converted into the standard ... more

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The given equation is converted into the standard ... more

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The given equation is converted into the standard ... more

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The given equation is converted into the standard ... more

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The given equation is converted into the standard ... more

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The given equation is converted into the standard ... more

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; The major axis is along the axis as the focus ... more

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; The major axis is along the axis as the focus ... more

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; The major axis is along the axis as the focus ... more

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; The major axis is parallel to the axis as the ... more

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; The major axis is parallel to the axis as the ... more

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; The major axis is parallel to the axis as the ... more

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; The given graph has the center at the origin ... more

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; The vertices of the ellipse on the major axis ... more

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; The vertices of the ellipse on the major axis ... more

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. ; The vertices of the ellipse on the major axis ... more

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. ; The vertices of the ellipse on the major axis ... more

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37.699. ; Comparing with the standard form of an ... more

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75.398. ; Comparing with the standard form of an ... more

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14.049. ; Comparing with the standard form of an ... more

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18.849. ; Rearrange the terms of equation to ... more

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28.274. ; Rearrange the terms of equation to ... more

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. ; The ellipse that can fit inside a box will ... more

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. ; The ellipse that can fit inside a box will ... more

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. ; Since the arch is in the shape of a semi-... more

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. ; Since the arch is in the shape of a ... more

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. ; Since the bridge is in the shape of a ... more

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. ; Considering the shape of the whispering galley... more

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. ; Considering the shape of the whispering ... more

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 Vertices: , Foci: ; The equation is compared ... more

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; Since the vertices and foci of the hyperbola ... more

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; Because the vertices and the foci are on a line... more

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Vertices: Co-vertices: Foci: Asymptotes: . ; Upon ... more

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Center: Vertices: and Co-vertices: and Foci: ... more

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. ; Calculate the value of using the information ... more

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A hyperbola is defined as a set of points such ... more

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 The center of the hyperbola is the origin. ; The ... more

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The foci of a hyperbola have the property that ... more

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The graph of the hyperbola would be symmetric ... more

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The center of the hyperbola is the midpoint of its... more

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The equation does not represent a hyperbola ... more

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. ; The equation represents a hyperbola, whose ... more

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It is not an equation of a hyperbola. ; The ... more

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. ; The equation represents a hyperbola. .The ... more

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. ; The equation represents a hyperbola. The ... more

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Equation: Vertices: Foci: Asymptotes: . ; The ... more

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Equation: Vertices: Foci: Asymptotes: .  ; Write ... more

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Equation: Vertices: .Foci: .  Asymptotes: .  ; The... more

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Equation: Vertices: Foci: .  Asymptotes: . ; The ... more

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Equation:  Vertices: and .   Foci: and .   ... more

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Equation:  Vertices: and  Foci: and  Asymptotes... more

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Equation:  Vertices: and  Foci: and  Asymptotes... more

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Equation:  Vertices: and  Foci: and  Asymptotes... more

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Vertices: Co-vertices: Foci: Asymptotes: . ; ... more

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Vertices: Co-vertices: Foci: Asymptotes: . ; ... more

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Vertices: Co-vertices: Foci: Asymptotes: . ; ... more

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Vertices: Co-vertices: Foci:Asymptotes: . ; ... more

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Vertices: Co-vertices:Foci: Asymptotes: . ; ... more

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Vertices:Co-vertices: Foci: Asymptotes: . ; ... more

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Vertices: Co-vertices: Foci: Asymptotes: . ; ... more

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. ; Compare the given equation with the standard ... more

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. ; Compare the given equation with the standard ... more

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. ; Compare the given equation with the standard ... more

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. ; Complete the squares in the given equation to... more

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. ; Complete the squares in the given equation to ... more

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The standard form of the hyperbola is expressed as... more

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The standard equation on the hyperbola is given by... more

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Compare the equation with the standard form of ... more

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Compare the equation with the standard form of ... more

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Compare the equation with the standard form of ... more

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The standard form of an equation of a hyperbola is... more

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The standard form of an equation of a hyperbola is... more

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The standard form of an equation of a hyperbola is... more

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Simplify the given expression to express it in ... more

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Simplify the given expression to express it in ... more

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Simplify the given expression to express it in ... more

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Simplify the given expression to express it in ... more

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Simplify the given expression to express it in ... more

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Simplify the given expression to express it in ... more

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; The vertices of the hyperbolas are and , and ... more

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; The vertices are and and the foci are at . As... more

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; The vertices are and , and the foci are . ... more

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  ; The center is , the vertices are and , and ... more

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; The center, vertices, and foci are , , and , ... more

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; The center, vertices, and foci are , , and , ... more

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; It is observed that , the center is (0,0), and... more

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; It is observed that the center is , the vertex ... more

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; It is observed from the graph that the center ... more

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; It is observed that the center is , the vertex... more

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; It is observed that the center (h,k) is (-3,-3... more

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; Solve the given equation for by following the ... more

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; Solve the given equation for by following the ... more

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and ; Simplify the given expression for in ... more

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Find in terms of from the equation of hyperbola... more

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Find in terms of from the equation of hyperbola... more

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; Here, the equation of asymptotes of hyperbola ... more

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; Here, the equation of asymptotes of hyperbola ... more

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; Here, the equation of asymptotes of hyperbola ... more

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; Here, the equation of the asymptotes of the ... more

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; Here, the equation of asymptotes of hyperbola ... more

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; Here, the equations of asymptotes of the ... more

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; Here, the equations of asymptotes of the ... more

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; Here, the equations of asymptotes of the ... more

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; Here, the equations of asymptotes of the ... more

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; Here, the equations of asymptotes of the ... more

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As the standard form of the parabola is . Compare ... more

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Corresponding to the given parabola, the standard ... more

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; The focus has the form , so the general ... more

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On comparing the given equation of the parabola ... more

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On comparing the given equation of parabola with ... more

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A parabola is the locus of all points in a plane ... more

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The graph will be open downward ; Suppose if the ... more

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The graph will be opened downwards ; Suppose if ... more

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If the value of is increases, then the graph is ... more

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The distance of focus and the directrix will ... more

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Equation is not a parabola. ; Find out the given ... more

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Equation is a parabola and standard form of it... more

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The given equation is not a parabola ; Check ... more

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; Check whether the given equation is a parabola... more

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; Check whether the given equation is a parabola... more

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  ; Rewrite the given equation in the standard ... more

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  ; Rewrite the given equation in the standard ... more

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  ; Rewrite the given equation in the standard ... more

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  ; Rewrite the given equation in the standard ... more

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; Rewrite the given equation in the standard ... more

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; Rewrite the given equation in the standard ... more

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  ; Rewrite the given equation in the standard ... more

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\\begin{aligned} \text{Vertex}  &=(-4,2) \\ \text{... more

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Vertex : Focus : Directrix: ; The left hand side ... more

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Vertex : Focus : Diretrix : ; The left hand side ... more

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Vertex :Focus :Directrix : ; The left hand side f ... more

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; The vertex is ; The focus is . ; The directrix ... more

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 The vertex is  . The focus is . The directrix is... more

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  ; Rewrite the given equation in the standard ... more

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The vertex is .The focus is .The directrix is . ; ... more

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 The vertex is . The focus is . The directrix is... more

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The vertex is .The focus is .The directrix is . ; ... more

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The vertex is .The focus is .The directrix is . ; ... more

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The vertex is .The focus is .The directrix is . ; ... more

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The vertex is  .The focus is .The directrix is... more

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The standard form of a parabola symmetric to the ... more

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The standard form of the parabola symmetric to the... more

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The standard form of a parabola symmetric to the ... more

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The standard form of a parabola symmetric to the ... more

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The standard form of a parabola symmetric to the ... more

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The standard form of a parabola symmetric to the ... more

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The standard form of a parabola symmetric to the ... more

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The standard form of a parabola symmetric to the ... more

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The standard form of a parabola symmetric to the ... more

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The standard form of a parabola symmetric to the ... more

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The standard form of a parabola symmetric to the ... more

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The general form of the equation of the parabola ... more

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The general form of the equation of the parabola ... more

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The general form of the equation of the parabola ... more

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; The vertex of the parabola is at , and the ... more

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; The vertex of the parabola is at , and the ... more

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; The vertex of the parabola is at , and the ... more

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; The vertex of the parabola is at , and the ... more

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; The vertex of the parabola is at , and the ... more

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; Calculate the equation of the parabola with the... more

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; The parabola in the first quadrant has its ... more

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; • The vertex of the parabola is given to be at... more

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; The vertex of the parabola is at , and the ... more

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(x+3)^2=\frac{-1}{16}(y-5) ; • The vertex of the ... more

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; • The vertex of the parabola is given to be at... more

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; • The vertex of the parabola is given to be at... more

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; • The vertex of the parabola is given to be at... more

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; • The vertex of the parabola given is and the ... more

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; • The vertex of the parabola is given to be at... more

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; • The vertex of the parabola is given to be at... more

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; The parabola , is of the form . Comparing the ... more

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; Find the equation of the parabola using the ... more

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above the vertex ; Assuming the center of the ... more

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above the vertex ; Assuming the center of the ... more

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; Assuming the center of the parabola as the ... more

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; First, convert inches to feet, so that the ... more

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and ; Assuming the center of parabola as the ... more

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and ; Assuming the center of the parabola as the ... more

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; Write the equation in the standard form by ... more

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; Write the equation in the standard form by ... more

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; On comparing with the general equation, , , . ... more

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The term causes a rotation of the graph to occur... more

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Parabola or line ; If the product of is zero, ... more

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Hyperbola ; If the discriminant of general ... more

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Ellipse ; Calculate the discriminant of the ... more

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Angle of rotation of axes ; When eliminating the ... more

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Ellipse. ; Calculate the discriminant of the ... more

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Parabola. ; Rewrite the equation in the form .... more

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Hyperbola. ; Rewrite the equation in the standard ... more

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Hyperbola. ; Rewrite the equation in the standard ... more

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Parabola. ; Rewrite the equation in the standard ... more

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Ellipse. ;  Rewrite the equation in the standard ... more

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Circle. ; Rewrite the equation in the standard ... more

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Circle. ; Rewrite the equation in the standard ... more

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Ellipse. ; Rewrite the equation in the standard ... more

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Ellipse. ; Rewrite the equation in the standard ... more

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Hyperbola. ; Rewrite the equation in the standard ... more

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Ellipse. ; Rewrite the equation in the standard ... more

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. ; Substitute and evaluate... more

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. ; Substitute and evaluate and... more

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. ; Substitute and evaluate and... more

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. ; Substitute and evaluate and... more

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. ; Substitute and evaluate and... more

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 ;   ; On comparing the equation with the ... more

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 ;   .  ; On comparing the equation with the ... more

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 ;  . ; On comparing the equation with the ... more

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 ;  . ; On comparing the equation with the ... more

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; On comparing with , the values of , and .... more

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; On comparing with , the values of , and .... more

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; On comparing with , the values of , and .... more

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; On comparing with , the values of , and .... more

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The new equation is . ; To find the new equation ... more

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; To find the new equation of the equation when ... more

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; To find the new equation of the equation or ... more

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; To find the new equation of the equation or ... more

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; To find the new equation of the equation when ... more

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; To find the new equation of the given ... more

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; To find the new equation of the given ... more

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; To find the new equation of the given ... more

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To find the new equation of the given expression... more

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To find the new equation of the given expression ... more

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To find the new equation of the given expression... more

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To find the new equation of the given expression... more

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To find the new equation of the given expression... more

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Calculate the and from the expression using the... more

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Calculate the and from the expression using the... more

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Calculate the and from the expression using the... more

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Compare the equation with to obtain the values ... more

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Compare the equation with to obtain the values ... more

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Compare the equation with to obtain the values ... more

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Compare the equation with to obtain the values ... more

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Compare the equation with to obtain the values ... more

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Compare the equation with to obtain the values ... more

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Compare the equation with to obtain the values ... more

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Compare the equation with to obtain the values ... more

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Compare the equation with to obtain the values ... more

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. ; Compare the equation with to obtain the ... more

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. ; Compare the equation with to obtain the ... more

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. ; Compare the equation with to obtain the ... more

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. ; Compare the given equation with the general ... more

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. ; Compare the given equation with the general ... more

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Ellipse; ; . ; The constant term in the ... more

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Substitute the value of and in to change the ... more

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. ; Since is given, so it is a cosine equation ... more

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. ; Unlike the polar equation, the rectangular ... more

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Eccentricity determines whether the conic is a ... more

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The denominator represents the angle and type of ... more

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The directrix is parallel to the polar axis. ; The... more

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The denominator of the polar form has . ; Since ... more

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The focus of the conic section is at the origin... more

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Hyperbola, , . ; As the constant term in the ... more

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Parabola, , . ; As the constant term in the ... more

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Ellipse, , . ; As the constant term in the ... more

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Hyperbola, , . ; As the constant term in the ... more

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Ellipse, , . ; As the constant term in the ... more

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Parabola, , . ; As the constant term in the ... more

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Parabola, , . ; As the constant term in the ... more

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Ellipse, , . ; As the constant term in the ... more

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Parabola, , . ; The polar equation can be written... more

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Hyperbola, , . ; The polar equation can be ... more

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Hyperbola, , ; The polar equation can be written ... more

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Hyperbola, , . ; The polar equation can be ... more

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; Unlike the polar equation, the rectangular ... more

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. ; Unlike the polar equation, the rectangular ... more

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. ; Unlike the polar equation, the rectangular ... more

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. ; Unlike the polar equation, the rectangular ... more

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. ; Unlike the polar equation, the rectangular ... more

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 . ; Given that the polar equation , change it ... more

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 . ; Given that the polar equation , change it ... more

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. ; Given that the polar equation , change it into... more

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 . ; Apply the distributive property to solve the ... more

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. ; Apply the distributive property to solve the ... more

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 . ; Apply the distributive property to solve the ... more

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. ; Simplify the given equation with the help of ... more

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. ; Simplify the given expression with the help of... more

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Divide the given equation by 2 in the numerator ... more

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Divide the given equation by 3 in the numerator ... more

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Divide the given equation by 5 in the numerator ... more

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Compare with the standard equation with the given... more

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Divide the given equation by 4 in the numerator ... more

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. ; Rewrite the given equation in the standard ... more

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Compare the given equation with the standard ... more

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Divide the given equation by 3 in the numerator ... more

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Divide the given equation by then compare with ... more

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Divide the given equation by then compare with ... more

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Divide the given equation by then compare with ... more

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. ; Rewrite the given equation in the standard ... more

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; Calculate the polar equation of the conic with... more

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; Calculate the polar equation of the cone with ... more

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; Calculate the polar equation of the cone with ... more

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  ; Calculate the polar equation of the cone with... more

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  ; Calculate the polar equation of the cone with... more

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  ; Calculate the polar equation of the cone with... more

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  ; Calculate the polar equation of the cone with... more

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; Calculate the polar equation of the cone  using... more

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; Calculate the polar equation of the cone with ... more

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; Calculate the polar equation of the cone with ... more

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; Calculate the polar equation of the cone with ... more

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  ; Calculate the polar equation of the cone with... more

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; Calculate the polar equation of the cone with ... more

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  ; Calculate the polar form of the given equation... more

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; Calculate the polar form of the given equation... more

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  ; Calculate the polar form of the given equation... more

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  ; Calculate the polar form of the given equation... more

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  ; Calculate the standard form of the given ... more

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  ; Calculate the standard form of the given ... more

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  ; Calculate the polar form of the given equation... more

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; Combine and terms and square the terms to get... more

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;Center: Vertices: Co-vertices: Foci: ; Make the ... more

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Compare with the standard form , where center is... more

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Compare with the standard form , where the center ... more

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Combine the terms and terms and square them to ... more

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Combine and terms and square them to write the ... more

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; The coordinates of the center and focus are ... more

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; The coordinates of the center and vertex are ... more

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; Distance of a foci from the ellipse is , where... more

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.Center: .Vertices: .Co-vertices: .Foci: . ; ... more

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Center: Vertices: Foci: ; Compare the equation ... more

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;Center: Vertices: Foci: ; Combine the terms and ... more

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Center: Vertices: Foci: ; Combine terms and and ... more

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Comparing with , where is the center. The center ... more

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Compare given equation with where center of ... more

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Combine and terms and square the respective ... more

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Combine the , terms and square the respective ... more

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. ; coordinates of the center and vertices are ... more

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; Vertex and foci have the same y coordinate. ... more

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Vertex: Focus: .Directrix: . ; From the given ... more

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, vertex at , focus at , directrix is ; ... more

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, vertex at , focus at , directrix is ; Rewrite ... more

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, vertex at , focus at , directrix is ; Rewrite ... more

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Rewrite the parabolic equation.Compared the ... more

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Rewrite the parabolic equation to standardise it.... more

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Rewrite the equation of the parabola to ... more

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Rewrite the equation of the parabola to ... more

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; The focus of the parabola is of the form and ... more

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; The directrix of the parabola has the form and... more

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At height from the vertex. ; Suppose the vertex ... more

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A parabola ; Compare the equation with the ... more

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A hyperbola ; Compare the equation to the general... more

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An ellipse ; Compare the equation to the general ... more

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, ; Compare the equation to the general equation ... more

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, ; Compare the equation to the general equation ... more

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Compare the equation to the general equation of a... more

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Compare the equation to the general equation of a... more

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Compare the equation to the general equation of a... more

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Eccentricity is 5, directrix is ; Since the ... more

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Eccentricity is , directrix is ; The constant term... more

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, ; In equation multiply numerator and ... more

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In equation , the denominator consist of with ... more

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In equation multiply numerator and denominator by... more

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In equation multiply numerator and denominator by... more

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In equation multiply numerator and denominator by... more

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; The directrix is , Since directrix is positive ... more

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; The directrix is , since the directrix is ... more

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Center=Vertices = Foci = ; equation of an ellipse... more

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this equations is in the form , major axis is ... more

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The  coordinate of the center .Vertices  A(-2,3+2\... more

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Center of the hyperbola .Vertices (0,-7)(0,-1) ; ... more

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 vertex .The standard form of the equation  focus ... more

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Circle. ; The given equation can be written in ... more

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Ellipse. ; Determine the type of conic section of ... more

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EllipseDirectrix is . ; Simplify the given ... more

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HyperbolaDirectrix is . ; Simplify the given ... more

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