Let P(a, b) be a point on the parabola y ^ 2 = 8x such that the...

Given $P\left(a, b\right)$ is point on ${y}^{2} = 8x$, such that tangent at $P$ pass through centre of ${x}^{2} + {y}^{2} - 10x - 14y + 65 = 0$ i.e. \left(5, 7\right)$</p><p>Now tangent at$P\left(2{t}^{2}, 4t\right)$will be$ty = x + 2{t}^{2}$</p><p>and it pass through \left(5, 7\right)$

So, $7t = 5 + 2{t}^{2}$

$⇒ t = 1, t = \dfrac{5}{2}$

So, point $P\left(2{t}^{2}, 4t\right) ⇒$\left(2, 4\right)$when$t = 1$</p><p>and \left(\dfrac{25}{2}, 10\right)$ when $t = \dfrac{5}{2}$

Now product of all possible value of $a$ will be $A = 2 \times \dfrac{25}{2} = 25$

And product of all possible value of $b$ will be $B = 4 \times 10 = 40$

$∴ A + B = 65$