In the standard (x, y) coordinate plane, an ellipse is centered at the origin (0, 0) and h… — ACT

In the standard $(x, y)$ coordinate plane, an ellipse is centered at the origin $(0, 0)$ and has vertices at $(-5, 0)$ and $(5, 0)$ . If the distance between the two foci of the ellipse is $8$ , what is the length of the minor axis of the ellipse?

Answer: 6 coordinate units

Answer

The correct answer is 6.

The correct answer is 6 because the ellipse has a horizontal major axis with

a = 5

and focal distance

c = 4

. Using the relationship

c^2 = a^2 - b^2

, we solve for the semi-minor axis

b

to get

b = 3

. The total length of the minor axis is

2b = 6

Step-by-Step Solution

Determine the semi-major axis length

a

a = 5

The vertices are at

(\pm 5, 0)

, which are

5

units from the center

(0, 0)

along the major axis.

Determine the distance from the center to each focus

c

c = 4

The distance between the two foci is

2c = 8

, so the distance from the center to a focus is

c = 4

Find the semi-minor axis length

b

b = 3

Using the relation

c^2 = a^2 - b^2

for ellipses, we get

4^2 = 5^2 - b^2 \implies b^2 = 9 \implies b = 3

Calculate the full length of the minor axis.

6

The length of the minor axis is

2b = 2(3) = 6

Key Concept

The relationship between the semi-major axis, semi-minor axis, and focal distance of an ellipse.

Estimated Time:1m 30s

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