Introduction to Ellipses

Show your work for each problem.

1.
What shape is an ellipse most similar to?
1. a)A triangle
2. b)A square
3. c)A straight line
4. d)A stretched circle
Answer: A stretched circle
An ellipse is essentially a stretched or compressed circle. When , an ellipse becomes a perfect circle.
2.
In the equation , what is ?
1. a)3
2. b)16
3. c)4
4. d)9
Answer: 16
By convention, is the larger value. Since , we have and . This means and .
3.
For the ellipse , what is the value of ?
Answer: 6
The larger denominator is 36, so . Taking the square root: .
4.
The formula for finding the distance from the center to each focus is:
1. a)
2. b)
3. c)
4. d)
Answer:
For ellipses: . The option is for hyperbolas, not ellipses.
5.
If and , calculate .
Answer: 4
, so .
6.
For the ellipse , the major axis is:
1. a)Circular
2. b)Diagonal
3. c)Horizontal
4. d)Vertical
Answer: Vertical
Since , we have under the term. This means the major axis is vertical (along the y-axis).
7.
Find the foci of the ellipse .
Answer: (-3, 0) and (3, 0)
1. What is (the larger denominator)? 25
2. What is ? 16
3. Calculate 9
4. What is ? 3
5. Is the major axis horizontal or vertical? horizontal
8.
What is the center of the ellipse ? Enter as (x, y).
Answer: (3, -2)
From , we get . From , we get . The center is .
9.
Find all vertices of the ellipse .
Answer: (-7, 0) and (7, 0)
1. What is ? 49
2. What is ? 7
3. Is the major axis horizontal or vertical? horizontal
4. The vertices are at . What are they? (-7, 0) and (7, 0)
10.
Which equation represents a vertical ellipse?
1. a)
2. b)
3. c)
4. d)
Answer:
In , the larger denominator (16) is under , so the major axis is vertical. The other two options with equal denominators represent circles, not ellipses.
11.
For the ellipse , what is the length of the major axis?
Answer: 20
The larger denominator is , so . The major axis length is .
12.
Write the equation of an ellipse centered at the origin with vertices at and co-vertices at .
Answer: x^2/36 + y^2/16 = 1
1. What is (distance to vertices)? 6
2. What is (distance to co-vertices)? 4
3. Is the major axis horizontal or vertical? horizontal
4. What is ? 36
5. What is ? 16
13.
For the ellipse , find the distance between the two foci.
Answer: 8
, . So , meaning . The distance between foci is .
14.
Find all key features of the ellipse : center, vertices, co-vertices, and foci.
Answer: Center: (-2, 1), Vertices: (-2, -4) and (-2, 6), Co-vertices: (-5, 1) and (1, 1), Foci: (-2, -3) and (-2, 5)
1. What is the center ? (-2, 1)
2. What is ? 25
3. Is the major axis horizontal or vertical? vertical
4. What is ? 5
5. What is ? 3
6. What is ? 4
15.
An ellipse has foci at and vertices at . What is ?
1. a)2
2. b)4
3. c)8
4. d)16
Answer: 4
From the foci, . From the vertices, . Using : , so and .
16.
Write the equation of an ellipse centered at with a vertical major axis of length 12 and a minor axis of length 8.
Answer: (x-1)^2/16 + (y+2)^2/36 = 1
1. What is (half the major axis)? 6
2. What is (half the minor axis)? 4
3. What is ? 36
4. What is ? 16
5. Since the major axis is vertical, goes under which term? y
17.
What happens to the foci when an ellipse becomes a circle?
1. a)They disappear completely
2. b)They move outside the curve
3. c)They become infinitely far apart
4. d)They merge into a single point (the center)
Answer: They merge into a single point (the center)
When (circle), , so . This means both foci coincide at the center of the circle.