Let P(a,b) be a point on the parabola y^2=8x such that the tangent at P passes through the…

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

Accepted Answer

Circle centre: $(5,7)$. Tangent to $y^2=8x$ at $(a,b)$: $by=4(x+a)$, with $b^2=8a$.

Through $(5,7)$: $7b=20+4a=20+\frac{b^2}2\Rightarrow b^2-14b+40=0\Rightarrow b=10	ext{ or }4$.

Corresponding $a=12.5$ or $2$. $A=12.5	imes2=25$, $B=10	imes4=40$.

$A+B=65.$

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