x = 4 1 2 ans: 2 = y 8 - berkeley city college. find the equation of the parabola with the given...
TRANSCRIPT
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
Ans: Center: (−1, 2); Transverse: y = 2; Vertices: (−2, 2), (0, 2);Foci (−1−
√3, 2), (−1 +
√3, 2); Asymptotes: y − 2 = ±
√2(x+ 1)
j. x2 + 8x− 3y2 − 6y + 4 = 0
Ans: Center: (−4,−1); Transverse: y = −1; Vertices: (−7,−1), (−1,−1);
Foci (−4− 2√3,−1), (−4 + 2
√3,−1); Asymptotes: y + 1 = ±
√3
3(x+ 4)