Let P (a, b) be a point on the parabola y² = 8x such that the tangent at P passes through the centre of the circle   x² + y² – 10x – 14y + 65 = 0. Let A be the product of all possible values of a and B be the product of all possible values of b. Then the value of A + B is equal to :

P(a, b) is point on y² = 8x, such that tangent at P pass through centre of x² + y² – 10x – 14y + 65 = 0 i.e. (5,7)

Tangent at P(at² , 2at) is ty = x + at²

A = 2 & it pass through (5, 7)

$\begin{array}{l}7\mathrm{t}=5+2{\mathrm{t}}^2\\ \Rightarrow \mathrm{t}=1, \mathrm{t}=\frac{5}{2}\\ \therefore \mathrm{P}\left({\mathrm{at}}^2, 2\mathrm{at}\right)\Rightarrow (2, 4)\text{ when }\mathrm{t}=1\\ \mathrm{\&}\left(\frac{25}{2}, 10\right)\text{ when }\mathrm{t}=\frac{5}{2}\\ \therefore \mathrm{A}=2\times \frac{25}{2}=25\\ \mathrm{B} = 4 \times 10 = 40 \therefore \mathrm{A}+\mathrm{B}=65 \end{array}$