Alg 3 (10) 15
Ellipse, Hyperbola
ELLIPSE DEF OF ELLIPSE
The ellipse is the set of all points in a plane such that the sum of the distances from two fixed points, (called the foci) is a constant.
PF + PF = 2a
Major axis =
Minor axis =
Sketch the graph of the following ellipses. Find the coordinates of the vertices and the foci.
1.
2.
ELLIPSES
3.
4.
5.
.
Day 2
TO FIND:
CENTER:
TO FIND “a”:
TO FIND “b”:
ORIENTATION:
1. Write the equation of the ellipse with
Center (1,1); Focus (1,3); Vertex (1,-9)
2. Foci (4,2) and (8,2) ; Major axis (MA) = (3,2), (9,2)
3. MA = (3,2) and (9,2); c = 3 (what is “c” ?)
4. Write the equation if the foci are (-8, 1), (8, 1) and the minor axis is 6.
5. Foci at ( 7, 0 ), ( -7, 0 ), and vertices at (8, 0 ) and ( -8, 0 )
Ellipses
(1) Sketch a graph of each of the following. Label the center, endpoints of the major and minor axes, and the focus points.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(2) Write the equation of the ellipse which satisfies each of the following.
(a) Foci at (-2 , 3) and (4 , 3) if the length of the major axis is 10.
(b) Foci at (-2 , 5) and (-2 , 1) if the length of the major axis is 8.
(c) Foci at (0 , 3) and (4 , 3) , vertices at (-4 , 3) and (8 , 3).
(d) Foci at (-2 , -3) and (-2 , 1) , vertices at (-2 , -6) and (-2 , 4).
(e) The endpoints of the major axis are (-4 , 5) and (2 , 5) , the endpoints of the minor axis are (-1 , 7) and (-1 , 3).
Answers
(1) (a) (e)
(b) (f)
(c) (g)
(d) (h)
(2) (a) (d)
(b) (e)
(c)
The hyperbola is the set of all points in a plane such that the difference of the distances from two fixed points, called foci, is constant.
Graph
`
1.
2.
Graph
3.
4. 5x2 - 10x - 4y2 - 16y = 31
HYPERBOLAS DAY 2 WRITING EQUATIONS
TO FIND:
CENTER:
TO FIND “a”:
TO FIND “b”:
ORIENTATION:
1. Center = (1,3); Transverse axis endpoint = (1,7); Focus = (1,-2)
2. TA endpoints = (3,-3), (-5,-3); slope of asymptotes
3. Foci = (1,0), (31,0); slope of asymptotes
4. Foci at (0, ±4), vertices at (0, ±2)
5. Vertices are ( -1, 3 ) and ( 5, 3 ) one focus is ( 7, 3 )
Hyperbolas
(1) Sketch a graph of each of the following. Label the center, foci and asymptotes.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(2) Write the equation of the hyperbola which satisfies each of the following.
(a) Foci are (-3 , 2) and (1 , 2) , the length of the transverse axis is 2
(b) Vertices are (-1 , 3) and (-1 , -1) , one focus is (-1 , 5)
(c) Vertices are (-6 , 2) and (0 , 2) , one focus is (2 , 2)
(d) Vertices are (-2 , -3) and (4 , -3) , slopes of the asymptotes are
(e) Foci are (1 , 8) and (1 , -2) , one vertex is (1 , 5)
Answers
(1) (a) (e)
(b) (f)
(c) (g)
(d) (h)
(2) (a) (d)
(b) (e)
(c)
Review Sheet
Ellipse and Hyperbolas
(1) Graph and find center, foci, and major and minor axis endpoints or asymptotes
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(3) Find the equation of the following:
a) Vertices (7,3) and (1,3); slope of asymptotes = 4/3
b) Focus (1,-5) and directrix y = 5
c) Foci (3,2) and (3,-6) ; length of Major Axis is 12.
d) Foci (14,9) and (-6,9) ; slope of asymptotes = 2
e) Asymptotes’ slopes= ±1 and whose foci are the endpoints of the horizontal diameter of the circle
f) Horizontal major axis of length 200, minor axis of length 22 and center at (6, -6)
g) Foci at (10,1) and (10,3) and a vertex at (10,11)
h) Vertical minor axis of length 12 and foci at (2, -6) and (-2, -6)
i) Foci at and slopes of asymptotes = ±2
Answers Ellipses and Hyperbolas
1a) b)
c) d)
e) f)
g) h)
2 a) b)
c) d)
e) f)
g) h)
i)
Extra Review Ellipses
1. Write the equation in standard form. Graph and label all important points.
a)
b)
c)
d)
e)
2. Write the equation of the ellipse whose center is the center of the circle given by
, one of the focus points is at (3, -1) and the length of the minor
axis is 4.
3. Write the equation of the line tangent to the circle at the point in the second
quadrant where x = -2.
4. Find the equation of the circle that has, as endpoints of a diameter (-3, 12) and (1, 10).
5. Find the equation of the ellipse with horizontal major axis of length 200, minor axis of
length 22 and center at (6, -6)
Answers
1. a)
center = (-4, 2) radius = points:
b)
center (6, -1), MA ep ma ep (6, 1), (6, -3) foci (7, -1), (5, -1)
c)
center (1, 1) radius 1 points (1, 2) (1, 0), (0, 1) (2, 1)
d)
center (0, 0) MA ep ma ep foci
e)
center radius , points
2.
3.
4.
5.