1. Find the equation of the parabola with the given information. Graph youranswer.
a. Focus at (0, 0); directrix x = 4
Ans: x− 2 = −18y2
b. Vertex at (0, 0); focus at (3, 0)
Ans: x =1
12y2
c. Focus at (3, 4); vertex at (3, 2)
Ans: y − 2 =1
8(x− 3)2
d. Vertex at (0, 0); directrix y = −4
Ans: y =1
16x2
2. Find the vertex, focus, directrix, and axis of symmetry of each parabola, andgraph your answer.
a. 4x = y2 − 4y
Ans: Vertex: (−1, 2); Focus: (0, 2);directrix: x = −2; axis of symmetry: y = 2
b. x2 = y + 2x
Ans: Vertex: (1,−1); Focus:(1,−3
4
);
directrix: y = −54; axis of symmetry: x = 1
c. y2 + 6y + 8x− 7 = 0
Ans: Vertex: (2,−3); Focus: (0,−3);directrix: x = 4; axis of symmetry: y = −3
d. x2 − 6x+ 10y − 1 = 0
Ans: Vetex: (3, 1); Focus:
(3,−3
2
);
directrix: y =7
2; axis of symmetry: x = 3
e. x2 + 10x− 2y + 21 = 0
Ans: Vertex: (−5,−2); Focus:(−5,−3
2
);
directrix: y = −52; axis of symmetry: x = −5
f. x2 + 8y + 4x− 4 = 0
Ans: Vertex: (−2, 1); Focus: (−2,−1);directrix: y = 3; axis of symmetry: x = −2
3. Find the equation of the circle with the given information, and graph youranswer:
a. Center (0, 0), radius 2.
Ans: x2 + y2 = 4
b. Center (0, 0), contains the point (−3,−4).Ans: x2 + y2 = 25
c. Center (−2,−4), contains the point (1,−1).
Ans: (x+ 2)2 + (y + 4)2 = 18
d. Diameter has endpoints (3, 4) and (−1, 2).Ans: (x− 1)2 + (y − 3)2 = 5
4. Find the center and radius, and graph the given circle:
a. x2 + y2 = 25
Ans: Center = (0, 0), r = 5
b. (x+ 2)2 + (y − 3)2 = 9
Ans: Center = (−2, 3), r = 3
c. x2 − 6x+ y2 = 1
Ans: Center = (3, 0), r =√10
d. x2 − x+ y2 + y =1
2
Ans: Center =
(1
2,−1
2
), r = 1
e. x2 +1
2x+ y2 +
1
2y =
1
8
Ans: Center = −(1
4,−1
4
), r =
1
2
5. Find the center, foci, and graph the given ellipse
a. 5x2 + y2 = 25
Ans: center: (0, 0); Foci: (0,−2√5), (0, 2
√5)
b. 4x2 + 3y2 = 48
Ans: center: (0, 0); Foci: (0,−2), (0, 2)
c. 9x2 + 4y2 = 9
Ans: Center: (0, 0); Foci:
(0,−√5
2
),
(0,
√5
2
)
d.(x+ 4)2
9+
(y + 2)2
4= 1
Ans: Center: (−4,−2); Foci: (−4−√5,−2), (−4 +
√5,−2)
e. 9(x− 3)2 + (y + 2)2 = 18
Ans: Center: (3,−2); Foci: (3,−6), (3, 2)
f. x2 + 3y2 − 12y + 9 = 0
Ans: Center: (0, 2); Foci: (−√2, 2), (
√2, 2)
g. 9x2 + 4y2 − 18x+ 16y − 11 = 0
Ans: Center: (1,−2); Foci: (1,−2−√5), (1,−2 +
√5)
h. 4x2 + y2 + 4y = 0
Ans: Center: (0,−2); Foci: (0,−2−√3), (0,−2 +
√3)
6. Find the equation of the ellipse with the given information. Graph youranswer.
a. Foci at (0,±2); length of major axis is 8
Ans:x2
12+
y2
16= 1
b. Focus at (−4, 0); vertices at (±5, 0)
Ans:x2
25+
y2
9= 1
c. Focus at (0,−4); vetrices at (0,±8)
Ans:x2
48+
y2
64= 1
d. Foci at (0,±3); x−intercepts are ±2
Ans:x2
4+
y2
13= 1
e. Vertices at (±4, 0); y−intercepts are ±1
Ans:x2
16+ y2 = 1
f. Center (−3, 1); vertex (−3, 3); focus (−3, 0)
Ans:(x+ 3)2
3+
(y − 1)2
4= 1
g. Foci at (1, 2) and (−3, 2); vertex at (−4, 2)
Ans:(x+ 1)2
9+
(y − 2)2
5= 1
h. Foci at (5, 1) and (−1, 1); length of the major axis is 8
Ans:(x− 2)2
16+
(y − 1)2
7= 1
i. Center at (1, 2); focus at (1, 4); contains the point (2, 2)
Ans: (x− 1)2 +(y − 2)2
5= 1
7. Find the equation of the hyperbola with the given information. Graph youranswer.
a. Center at (0, 0); focus at (3, 0); vertex at (1, 0)
Ans: x2 − y2
8= 1
b. Focus at (0, 6); vertices at (0,−2) and (0, 2)
Ans: −x2
32+
y2
4= 1
c. Vetices at (−4, 0) and (4, 0); asymptote y = 2x
Ans:x2
16− y2
64= 1
d. Foci at (−4, 0) and (4, 0); asymptote y = −x
Ans:x2
8− y2
8= 1
e. Center at (4,−1); focus at (7,−1); vertex at (6,−1)
Ans:(x− 4)2
4− (y + 1)2
5= 1
f. Center at (−3, 1); focus at (−3, 6); vetex at (−3, 4)
Ans: −(x+ 3)2
16+
(y − 1)2
9= 1
g. Focus at (−4, 0); vertices at (−4, 4) and (−4, 2)
Ans: −(x+ 4)2
8+ (y − 3)2 = 1
h. Vertices at (1,−3) and (1, 1); asymptote y + 1 =3
2(x− 1)
Ans: −9(x− 1)2
16+
(y + 1)2
4= 1
8. Find the center, transverse axis, vertices, foci, and asymptotes. Graph theequation.
a.x2
25− y2
9= 1
Ans: Center: (0, 0); Transverse: y = 0; Vertices: (−5, 0), (5, 0);
Foci (−√34, 0), (
√34, 0); Asymptotes: y = ±3
5x
b. −x2
4+
y2
16= 1
Ans: Center: (0, 0); Transverse: x = 0; Vertices: (0,−4), (0, 4);Foci (0,−2
√5), (0, 2
√5); Asymptotes: y = ±2x
c. −x2 + 4y2 = 16
Ans: Center: (0, 0); Transverse: x = 0; Vertices: (0,−2), (0, 2);
Foci (0,−2√5), (0, 2
√5); Asymptotes: y = ±1
2x
d.(x− 2)2
4− (y + 3)2
9= 1
Ans: Center: (2,−3); Transverse: y = −3; Vertices: (0,−3), (4,−3);
Foci (2−√13,−3), (2 +
√13,−3); Asymptotes: y + 3 = ±3
2(x− 2)
e. −(x+ 2)2 +(y − 2)2
4= 1
Ans: Center: (−2, 2); Transverse: x = −2; Vertices: (−2, 0), (−2, 4);Foci (−2, 2−
√5), (−2, 2 +
√5); Asymptotes: y − 2 = ±2(x+ 2)
f.(x+ 1)2
4− (y + 2)2
4= 1
Ans: Center: (−1,−2); Transverse: y = −2; Vertices: (−3,−2), (1,−2);Foci (−1− 2
√2,−2), (−1 + 2
√2,−2); Asymptotes: y + 2 = ±(x+ 1)
g. x2 − y2 − 2x− 2y − 1 = 0
Ans: Center: (1,−1); Transverse: y = −1; Vertices: (0,−1), (2,−1);Foci (1−
√2,−1), (1 +
√2,−1); Asymptotes: y + 1 = ±(x− 1)
h. −4x2 − 8x+ y2 − 4y − 4 = 0
Ans: Center: (−1, 2); Transverse: x = −1; Vertices: (−1, 0), (−1, 4);Foci (−1, 2−
√5), (−1, 2 +
√5); Asymptotes: y − 2 = ±2(x+ 1)
i. 2x2 + 4x− y2 + 4y − 4 = 0