Let PA and PB are two tangents drawn from point P, outside the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , (a > b). If $H_{1}$ and $H_{2}$ are reflections of $F_{1}$ and $F_{2}$ respectively ( $F_{1}, F_{2}$ are the foci of the ellipse) in the tangents PA and PB, then which of the following option(s) are correct

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$Δ P F_{2} H_{1} a n d Δ P F_{1} H_{2}$ are similar

$Δ P F_{2} H_{1} a n d Δ P F_{1} H_{2}$ are congruent

$∠ P H_{2} F_{1} = ∠ P F_{2} H_{1}$

mid-points of line joining $F_{1} H_{1}$ lies on a circle of radius a

answer is A, B, C, D.

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Detailed Solution

By reflection property of ellipse $∠ P A F_{2} = ∠ Q A F_{1}$
$∵ H_{1}$ is reflection of F1 $∴ F_{1} ∠ Q A H_{1} = ∠ Q A F_{1}$
Now $P H_{2} = P F_{2}, P H_{1} = P F_{1}$
and $F_{2} H_{1} = F_{2} A + A F_{1} = 2 a$
and $F_{1} H_{2} = F_{1} B + B H_{2} = F_{1} B + F_{2} B = 2 a$