First slide

Ellipse

Question

If the eccentric angles of two points P and Q on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}$ are $α, β$ such that $α + β = \frac{π}{2}$ then,the locus of the point of intersection of the normals at $P$ and $Q$ is

Moderate

A

$a x + b y = 0$
B

$a x - b y = 0$
C

$x + y = 0$
D

$x + y = a + b$

Solution

Equations of the normal at $P$ and $Q$ are
$a x \sec α -$ by $cosec α = a^{2} - b^{2}$
$a x \sec β -$ by $cosec β = a^{2} - b^{2}$
If they intersect at $(h, k) .$
$a h \sec α - b k cosec α = a h cosec α - b k \sec α$
$(∵ β = π / 2 - α)$
$(a h + b k) (\sec α - cosec α) = 0 \Rightarrow a h + b k = 0$
Locus of $(h, k)$ is $a x + b y = 0$

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