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Q.

Let $A B : y = x$ be a chord to the parabola $y^{2} + 4 x = 4$ and $P$ be any variable point on the ellipse $x^{2} + 9 (y - 4)^{2} - 9 = 0 .$ $If Δ_{1} and Δ_{2}$ be the minimum and maximum area of triangle $A B P$ respectively, then

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a

$Δ_{2} = 2 (4 + \sqrt{10})$

b

$Δ_{2} - Δ_{1} = 8 \sqrt{5}$

c

$\frac{Δ_{2}}{Δ_{1}} = \frac{13 + 4 \sqrt{10}}{3}$

d

$\frac{Δ_{2}}{Δ_{1}} = \frac{13 + 4 \sqrt{10}}{6}$

answer is B, C.

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$\begin{aligned} \Rightarrow Δ_{1} = 2 \sqrt{2} (4 - \sqrt{10}) \\ and Δ_{2} = 2 \sqrt{2} (4 + \sqrt{10}) \end{aligned}$

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Let AB:y=x be a chord to the parabola y2+4x=4 and P be any variable point on the ellipse x2+9(y−4)2−9=0. If Δ1 and Δ2 be the minimum and maximum area of triangle ABP respectively, then