Question

Let $P ( a , b )$ be a point on the parabola $y ^2=8 x$ such that the tangent at $P$ passes through the centre of the circle $x^2+y^2-10 x-14 y+65=0$. 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 $y ^2=8 x$, such that tangent at $P$ pass through centre of $x^2+y^2-10 x-$ $14 y +65=0$ i.e. $(5,7)$
Tangent at $P \left( at ^2, 2 at \right)$ is ty $= x + at ^2$
$A =2 \&$ it pass through $(5,7)$
$7 t=5+2 t^2$
$ \Rightarrow t=1, t=\frac{5}{2}$
$ \therefore P\left(a t^2, 2 a t\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 $
$B =4 \times 10=40 \therefore A+B=65$