To the parabola $x^{2} = 4 a y$ tangent is drawn at any point $(x_{1}, y_{1})$ other than the vertex. If tangents are drawn to the circle $x^{2} + y^{2} = 4 a^{2}$ from any point on this tangent such that all the corresponding chords of contact corresponding pass through a fixed point $(x_{2}, y_{2})$ then

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$\frac{x_{1}}{2}, 2 a, x_{2}$ are in Geometric Progression

$x_{1} x_{2} + y_{1} y_{2} = 4 a^{2}$

$y_{1}, 2 a, y_{2}$ are in Geometric Progression

$- 4, \frac{x_{1}}{x_{2}}, \frac{y_{1}}{y_{2}}$ are in Geometric Progression

answer is C, D.

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Detailed Solution

Consider $(x_{1}, y_{1}) \equiv (2 a t, a t^{2}) \Rightarrow Equation of tangent \equiv t x = y + a t^{2}$
$\Rightarrow COC w.r.t (λ, t λ - a t^{2}) is λ (x + t y) - (a t^{2} y + 4 a^{2}) = 0$
$\Rightarrow (x_{2}, y_{2}) \equiv (\frac{4 a}{t}, - \frac{4 a}{t^{2}})$