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( y 1 ) 2 49 ( x + 1 ) 2 4 = 1


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x 2 4 y 2 + 6 x + 32 y 91 = 0

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2 y 2 x 2 12 y 6 = 0


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For the following exercises, find the equation of the hyperbola.

Center at ( 0 , 0 ) , vertex at ( 0 , 4 ) , focus at ( 0 , −6 )

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Foci at ( 3 , 7 ) and ( 7 , 7 ) , vertex at ( 6 , 7 )

( x 5 ) 2 1 ( y 7 ) 2 3 = 1

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The Parabola

For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

( x + 2 ) 2 = 1 2 ( y 1 )

( x + 2 ) 2 = 1 2 ( y 1 ) ; vertex: ( −2 , 1 ) ; focus: ( −2 , 9 8 ) ; directrix: y = 7 8

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y 2 6 y 6 x 3 = 0

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x 2 + 10 x y + 23 = 0

( x + 5 ) 2 = ( y + 2 ) ; vertex: ( 5 , 2 ) ; focus: ( 5 , 7 4 ) ; directrix: y = 9 4

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For the following exercises, graph the parabola, labeling vertex, focus, and directrix.

( y 1 ) 2 = 1 2 ( x + 3 )


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x 2 8 x 10 y + 46 = 0

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2 y 2 + 12 y + 6 x + 15 = 0


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For the following exercises, write the equation of the parabola using the given information.

Focus at ( −4 , 0 ) ; directrix is x = 4

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Focus at ( 2 , 9 8 ) ; directrix is y = 7 8

( x 2 ) 2 = ( 1 2 ) ( y 1 )

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A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.

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Rotation of Axes

For the following exercises, determine which of the conic sections is represented.

16 x 2 + 24 x y + 9 y 2 + 24 x 60 y 60 = 0

B 2 4 A C = 0 , parabola

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4 x 2 + 14 x y + 5 y 2 + 18 x 6 y + 30 = 0

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4 x 2 + x y + 2 y 2 + 8 x 26 y + 9 = 0

B 2 4 A C = 31 < 0 , ellipse

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For the following exercises, determine the angle θ that will eliminate the x y term, and write the corresponding equation without the x y term.

x 2 + 4 x y 2 y 2 6 = 0

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x 2 x y + y 2 6 = 0

θ = 45 , x 2 + 3 y 2 12 = 0

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For the following exercises, graph the equation relative to the x y system in which the equation has no x y term.

9 x 2 24 x y + 16 y 2 80 x 60 y + 100 = 0

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x 2 x y + y 2 2 = 0

θ = 45

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6 x 2 + 24 x y y 2 12 x + 26 y + 11 = 0

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Conic Sections in Polar Coordinates

For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix.

r = 10 1 5   cos   θ

Hyperbola with e = 5 and directrix 2 units to the left of the pole.

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r = 6 3 + 2   cos   θ

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r = 1 4 + 3   sin   θ

Ellipse with e = 3 4 and directrix 1 3 unit above the pole.

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r = 3 5 5   sin   θ

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For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci.

r = 8 4 + 3   sin   θ

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r = 10 4 + 5   cos   θ


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r = 9 3 6   cos   θ

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For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.

Directrix is x = 3 and eccentricity e = 1

r = 3 1 + cos     θ

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Directrix is y = −2 and eccentricity e = 4

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Practice test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.

x 2 9 + y 2 4 = 1

x 2 3 2 + y 2 2 2 = 1 ; center: ( 0 , 0 ) ; vertices: ( 3 , 0 ) , ( –3 , 0 ) , ( 0 , 2 ) , ( 0 , −2 ) ; foci: ( 5 , 0 ) , ( 5 , 0 )

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9 y 2 + 16 x 2 36 y + 32 x 92 = 0

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For the following exercises, sketch the graph, identifying the center, vertices, and foci.

( x 3 ) 2 64 + ( y 2 ) 2 36 = 1

center: ( 3 , 2 ) ; vertices: ( 11 , 2 ) , ( −5 , 2 ) , ( 3 , 8 ) , ( 3 , −4 ) ; foci: ( 3 + 2 7 , 2 ) , ( 3 2 7 , 2 )

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2 x 2 + y 2 + 8 x 6 y 7 = 0

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Write the standard form equation of an ellipse with a center at ( 1 , 2 ) , vertex at ( 7 , 2 ) , and focus at ( 4 , 2 ).

( x 1 ) 2 36 + ( y 2 ) 2 27 = 1

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A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?

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For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.

x 2 49 y 2 81 = 1

x 2 7 2 y 2 9 2 = 1 ; center: ( 0 , 0 ) ; vertices ( 7 , 0 ) , ( −7 , 0 ) ; foci: ( 130 , 0 ) , ( 130 , 0 ) ; asymptotes: y = ± 9 7 x

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16 y 2 9 x 2 + 128 y + 112 = 0

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For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.

( x 3 ) 2 25 ( y + 3 ) 2 1 = 1

center: ( 3 , −3 ) ; vertices: ( 8 , −3 ) , ( −2 , −3 ) ; foci: ( 3 + 26 , −3 ) , ( 3 26 , −3 ) ; asymptotes: y = ± 1 5 ( x 3 ) 3

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y 2 x 2 + 4 y 4 x 18 = 0

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Write the standard form equation of a hyperbola with foci at ( 1 , 0 ) and ( 1 , 6 ) , and a vertex at ( 1 , 2 ) .

( y 3 ) 2 1 ( x 1 ) 2 8 = 1

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For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.

3 x 2 12 x y + 11 = 0

( x 2 ) 2 = 1 3 ( y + 1 ) ; vertex: ( 2 , −1 ) ; focus: ( 2 , 11 12 ) ; directrix: y = 13 12

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For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.

( x 1 ) 2 = −4 ( y + 3 )

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y 2 + 8 x 8 y + 40 = 0


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Write the equation of a parabola with a focus at ( 2 , 3 ) and directrix y = −1.

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A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

Approximately 8.49 feet

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For the following exercises, determine which conic section is represented by the given equation, and then determine the angle θ that will eliminate the x y term.

3 x 2 2 x y + 3 y 2 = 4

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x 2 + 4 x y + 4 y 2 + 6 x 8 y = 0

parabola; θ 63.4

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For the following exercises, rewrite in the x y system without the x y term, and graph the rotated graph.

11 x 2 + 10 3 x y + y 2 = 4

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16 x 2 + 24 x y + 9 y 2 125 x = 0

x 2 4 x + 3 y = 0

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For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.

r = 5 4 + 6   cos   θ

Hyperbola with e = 3 2 , and directrix 5 6 units to the right of the pole.

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For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci.

r = 12 4 8   sin   θ

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r = 2 4 + 4   sin   θ

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Find a polar equation of the conic with focus at the origin, eccentricity of e = 2 , and directrix: x = 3.

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Questions & Answers

write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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