Conics Final Review: Show all work on a separate sheet of paper.
1) Write an equation in standard form for the hyperbola with center 0,0 , vertex (0,12), and
focus (0,13)
2) Write an equation in standard form for the hyperbola with center(−2,3), vertex (−2,8), and
focus (−2,9)
3) Engineers are building semi-elliptical pedestrian bridges across two highways. The larger
bridge will be 1.5 times as wide and three times as tall as the smaller bridge. The equation for the
bridge over the smaller highway is
x2
y2

 1 , measured in feet. Find the dimensions of the
100 144
larger bridge. Then, find an equation for the design of the larger bridge.
4) Engineers are building semi-elliptical pedestrian bridges across two highways. The larger
bridge will be 1/3 times as wide and half as tall as the smaller bridge. The equation for the bridge
x2
y2
over the smaller highway is

 1 , measured in feet. Find the dimensions of the larger
900 64
bridge. Then, find an equation for the design of the larger bridge.
5) Identify the conic section that the equation 4 x 2  10 xy  3 y 2  8x  6 y  9  0 represents.
6) Identify the conic section that the equation 2 x 2  2 y 2  8x  6 y  9  0 represents.
7) Identify the conic section that the equation 2 x 2  10 xy  6 y 2  8x  6 y  9  0 represents.
8) Identify the conic section that the equation 2 x 2  8 x  6 y  9  0 represents.
9) Write the equation in standard form for the parabola with vertex 0,0 and directix y  2 .
10) Write the equation in standard form for the parabola with vertex 2,4 and directix x  1 .
11) Complete the square 2 x 2  12 x  15  0
12) Complete the square 3 x 2  12 x  1  0
13) Complete the square x 2  9 x  2  0
14) An airplane makes a dive that can be modeled by the equation
 9 x 2  16 y 2  36 x  32 y  146  0 , measured in hundreds of feet, with the ground
represented by the x-axis. How close to the ground does the airplane pass?
15) Find the center and radius of the circle with the equation  x  4   y  5  81
2
16) Write the equation of a circle with center (−3,6) and radius 7.
2
17) Graph the ellipse
18) Graph the ellipse
(𝑥+3)2
81
(𝑥−2)2
16
+
+
(𝑦−7)2
25
(𝑦+1)2
36
=1
=1
19) Find the vertecies, co-vertecies, and asymptotes of the hyperbola
20) Find the vertecies, co-vertecies, and asymptotes of the hyperbola
(𝑥+3)2
9
(𝑦+2)2
144
−
−
(𝑦−1)2
16
(𝑥−1)2
25
= 1. Graph.
= 1. Graph.
21) Find the vertex, value of p, axis of symmetry, focus, and directrix of the parabola: 𝑦 + 3 =
1
(𝑥 − 2)2 . Graph
12
22) Find the vertex, value of p, axis of symmetry, focus, and directrix of the parabola: 𝑥 − 1 =
1
(𝑦 − 5)2 . Graph
2
Solutions:
𝑦2
𝑥2
18) Center: (2,-1)
Vertecies: (2,5), (2,-7)
Co-Vertecies: (-2,-1), (6,-1)
1) 144 − 25 = 1
2)
(𝑦−3)2
25
−
(𝑥+2)2
11
=1
𝑥2
𝑦2
3) 30ft wide, 72ft high 225 + 1296 = 1
4) 20ft wide, 8ft high
5) Ellipse
6) Circle
7) Hyperbola
8) Parabola
1
9) 𝑦 = 8 𝑥 2
𝑥2
100
+
𝑦2
16
=1
1
10) 𝑥 − 2 = 12 (𝑦 + 4)2
11) (𝑥 − 3)2 =
2
12) (𝑥 − 2) =
9 2
)
2
33
2
13
3
73
13) (𝑥 +
= 4
14) 200ft
15) (4, -5) r = 9
16) (𝑥 + 3)2 + (𝑦 − 6)2 = 49
17) Center (-3,7)
Vertecies: (-12,7), (6,7)
Co-Vertecies: (-3,12) (-3,2)
19) Center (-3,1)
Vertecies: (-6,1), (0,1)
Co- Vertecies: (-3,5), (-3,-3)
4
Asymptotes: 𝑦 = 3 𝑥 + 5,
−4
𝑦= 3 𝑥−3
20) Center (1,-2)
Vertecies: (1,10), (1,-14)
Co- Vertecies: (-4,-2), (6,-2)
12
22
Asymptotes: 𝑦 = 5 𝑥 − 5 ,
𝑦=
21) p = 3
AoS: x = 2
Focus: (2,0)
Directrix: y = -6
22) p = 0.5
AoS: y = 5
Focus: (1.5,5)
Directrix: x = 0.5
−12
5
2
𝑥+5