Ellipses in Standard Form

Show your work for each problem.

1.
What is the center of the ellipse ?
1. a)
2. b)
3. c)
4. d)
Answer:
From , we get . From , we get . The center is .
2.
Is the ellipse horizontal or vertical?
1. a)Vertical
2. b)Neither
3. c)Horizontal
4. d)Both
Answer: Vertical
Since , the larger denominator is under , so is associated with . This means the major axis is vertical.
3.
For the ellipse , what is the value of ?
Answer: 6
The larger denominator is , so and .
4.
For an ellipse with and , find (the distance from center to focus).
Answer: 4
, so .
5.
What are the vertices of the ellipse ?
1. a)
2. b)
3. c)
4. d)
Answer:
Since , so . The major axis is horizontal (larger denominator under ), so vertices are at .
6.
Find the value of for the ellipse .
Answer: 6
, . So , and .
7.
Find the foci of the ellipse .
Answer: (-3, 2) and (5, 2)
1. What is the center ? (1, 2)
2. What is (the larger denominator)? 25
3. Calculate 16
4. What is ? 4
5. Since the major axis is horizontal, where are the foci? (1-4, 2) and (1+4, 2)
8.
Write the equation in standard form for an ellipse with center , vertices at , and co-vertices at .
Answer: x^2/64 + y^2/25 = 1
1. What is (distance from center to vertex)? 8
2. What is (distance from center to co-vertex)? 5
3. Is this ellipse horizontal or vertical? horizontal
4. What is ? 64
5. What is ? 25
9.
The foci of the ellipse are located at:
1. a)
2. b)
3. c)
4. d)
Answer:
(larger), . So , . Since is under , the major axis is vertical, so foci are at .
10.
An ellipse has foci at and vertices at . What is ?
Answer: 3
, . So , and .
11.
Convert to standard form:
Answer: (x-2)^2/16 + (y+1)^2/36 = 1
1. Group the x terms and y terms (9x^2 - 36x) + (4y^2 + 8y) = 104
2. Factor out the coefficients 9(x^2 - 4x) + 4(y^2 + 2y) = 104
3. Complete the square for x: what value completes ? 4
4. Complete the square for y: what value completes ? 1
5. What is the new right side after adding to both sides? 144
6. Divide both sides by 144 to get standard form (x-2)^2/16 + (y+1)^2/36 = 1
12.
Find the equation in standard form for an ellipse with foci at and major axis length of 10.
Answer: x^2/9 + y^2/25 = 1
1. Is the major axis horizontal or vertical? vertical
2. What is (distance from center to focus)? 4
3. What is (half the major axis length)? 5
4. Calculate 9
5. Write the standard form (vertical ellipse centered at origin) x^2/9 + y^2/25 = 1
13.
What is the eccentricity of the ellipse ? Give your answer as a fraction.
Answer: 5/13
, . , so and . Eccentricity .
14.
Which ellipse has the highest eccentricity?
1. a)
2. b)
3. c)
4. d)
Answer:
Option A: . Option B: . Option C: . Option D: . The first ellipse has the highest eccentricity.
15.
Find all key features of the ellipse : center, vertices, co-vertices, and foci.
Answer: Center: (-3, 1); Vertices: (-9, 1) and (3, 1); Co-vertices: (-3, 1-sqrt(20)) and (-3, 1+sqrt(20)); Foci: (-7, 1) and (1, 1)
1. What is the center ? (-3, 1)
2. What is ? 36
3. What is ? 6
4. Is the major axis horizontal or vertical? horizontal
5. What is ? 16
6. What is ? 4
16.
For the ellipse , what is the value of ?
Answer: 7
The smaller denominator is , so and .
17.
Write the equation of an ellipse centered at with (horizontal) and .
Answer: (x-2)^2/16 + (y+1)^2/9 = 1
1. What is for center at ? (x-2)^2
2. What is for center at ? (y+1)^2
3. What is ? 16
4. What is ? 9
5. Write the complete equation (x-2)^2/16 + (y+1)^2/9 = 1