Answer the following by appropriately matching the lists based on the information given in the p....

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Answer the following by appropriately matching the lists based on the information given in the paragraphLet the circles \(C_1: x^2+y^2=9\) and \(C_2:(x-3)^3+(y-4)^2=16\), 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 expression given in the List-I whose values are given in List-II below :
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