Q. An ellipse has its center at (1, −2), one focus at (3, −2) and one vertex at (5, −2). Then the length of its latus rectum is :

Correct Answer: 6

Explanation

The center of the ellipse is

(h, k) = (1, −2)

The given focus is

(3, −2)

Since the focus and center lie on the same horizontal line, the major axis is along the x-axis.

Distance of the focus from the center gives

c = |3 − 1| = 2

The given vertex is

(5, −2)

Distance of the vertex from the center gives

a = |5 − 1| = 4

For an ellipse,

c² = a² − b²

Substituting values,

2² = 4² − b²

4 = 16 − b²

b² = 12

b = 2√3

Length of the latus rectum of an ellipse is given by

Length = 2b² / a

Substituting values,

Length = 2 × 12 / 4 = 6

Hence, the required length of the latus rectum is

6

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