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Let $T_{1} a n d T_{2}$ be two distinct common tangents to the ellipse $E \equiv x^{2} + 2 y^{2} + 3 = 2 x + 8 y$ and parabola $P \equiv y^{2} + 16 = 4 y + 12 x .$ Suppose $T_{1}$ touches E and P at $E_{1} a n d P_{1}$ respectively. Suppose $T_{2}$ touches E and P at $E_{2} a n d P_{2}$ respectively. If slope of $T_{1}$ is greater than that of $T_{2}$ , then which of the following options is/are correct?

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The area of quadrilateral $E_{1} P_{1} P_{2} E_{1}$ equals 35 square units.

Point of intersection of $T_{1} a n d T_{2}$ is at 5 units from $(1, - 2)$

$E_{1} P_{2} + E_{2} P_{1} = 2 \sqrt{74}$

The area of $Δ E_{1} P_{1} E_{2}$ equals 5 square units.

answer is A, B, C, D.

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