Question

Let $P(a,b)$ be a point on the parabola $y^2=8x$ such that the tangent at $P$ passes through the centre of the circle $x^2+y^2-10x-14y+65=0$. 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 :

• A. 0
• B. 25
• C. 40
• D. 65

Answer 1

Answer: (D)

$P(a,b)$ 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. $(5,7)$
Tangent at $P\left(a{t}^2,2at\right)$ is ty $=x+a{t}^2$
$A=2\&$ it pass through $(5,7)$
$7t=5+2t^2$
$\Rightarrow t=1,t=\frac{5}{2}$
$\therefore P\left(a{t}^2,2at\right)\Rightarrow (2,4)\text{ when }t=1$
$\left(\frac{25}{2},10\right)\text{ when }t=\frac{5}{2}$
$\therefore A=2\times \frac{25}{2}=25$
$B=4\times 10=40\therefore A+B=65$