JEE Advanced 2019 · Paper 2, Mathematics question 17

Answer Q.17 and Q.18 by appropriately matching the lists based on the information given in the paragraph.

Let the circles $C_1: x^2 + y^2 = 9$ and $C_2: (x-3)^2 + (y-4)^2 = 16$, intersect at the points $X$ and $Y$. Suppose that another circle $C_3: (x-h)^2 + (y-k)^2 = r^2$ satisfies the following conditions:

(i) centre of $C_3$ is collinear with the centres of $C_1$ and $C_2$,

(ii) $C_1$ and $C_2$ both lie inside $C_3$, and

(iii) $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$.

Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be a tangent to the parabola $x^2 = 8\alpha y$.

There are some expressions given in the List-I whose values are given in List-II below: