CAT (QA) PYQs

Let $C$ be the circle $x^2 + y^2 + 4x - 6y - 3 = 0$ and $L$ be the locus of the point of intersection of a pair of tangents to $C$ with the angle between the two tangents equal to $60°$. Then, the point at which $L$ touches the line $x = 6$ is

Let $C$ be a circle of radius $5$ meters having center at $O$. Let $P Q$ be a chord of $C$ that passes through points $A$ and $B$ where $A$ is located $4$ meters north of $O$ and $B$ is located $3$ meters east of $O$. Then, the length of $P Q$, in meters, is nearest to

On a rectangular metal sheet of area $135 \mathrm{sq}$ in, a circle is painted such that the circle touches two opposite sides. If the are the sheet left unpainted is two-thirds of the painted area then the perimeter of the rectangle in inches is