Straight Lines · JEE Advanced previous year questions

Let $S$ denote the locus of the point of intersection of the pair of lines $$4x-3y=12\alpha ,$$ $$4\alpha x+3\alpha y=12,$$ where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $(p,0)$ and $(0,q)$, $q>0$, and parallel to the line $4x-\frac{3}{\sqrt{2}}y=0$. Then the value of $pq$ is

1. A.$-6\sqrt{2}$
2. B.$-3\sqrt{2}$
3. C.$-9\sqrt{2}$
4. D.$-12\sqrt{2}$

A straight line drawn from the point $P(1,3,2)$, parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$, intersects the plane $L_1:x-y+3z=6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L_1$ intersects the plane $L_2:2x-y+z=-4$ at the point $R$. Then which of the following statements is (are) TRUE?

1. A.The length of the line segment $PQ$ is $\sqrt{6}$
2. B.The coordinates of $R$ are $(1,6,3)$
3. C.The centroid of the triangle $PQR$ is $\left(\frac{4}{3},\frac{14}{3},\frac{5}{3}\right)$
4. D.The perimeter of the triangle $PQR$ is $\sqrt{2}+\sqrt{6}+\sqrt{11}$