First slide

Ellipse

Question

The major axis and minor axis of an ellipse are respectively, x - 2y - 5 - 0 and 2x + y + 10 = 0. If one end of latus rectum is (3 ,4), then the foci are

Moderate

A

(5, 0)
B

(-7, -6)
C

(-11, -8)
D

(11, 3)

Solution

Foot of perpendicular from one end of latus rectum L(3, 4) on major axis which is focus is given by
$\frac{x - 3}{1} = \frac{y - 4}{- 2} = \frac{- (3 - 2 (4) - 5)}{1^{2} + (- 2)^{2}} = 2$
$∴ S (x, y) \equiv S (5, 0)$
Another focus is image of s in the line 2x + y + 10 = 0.
$\begin{array}{l} \frac{x - 5}{2} = \frac{y - 0}{1} = \frac{- 2 (2 (5) + 0 + 10)}{2^{2} + 1^{2}} = - 8 \\ ∴ S^{'} (x, y) \equiv S (- 11, - 8) \end{array}$

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