Question

Let $P,Q$ be the end points of the chord of contact of the point $R(2,5)$ with respect to $y^2=8x.$ The length of the intercept made by the circle with $PQ$ as a diameter, on the $y-$axis is equal to

• A. $0$
• B. $2\sqrt{5}$
• C. $4\sqrt{5}$
• D. $10$

Answer 1

The correct option is B $2\sqrt{5}$

Given parabola $:y^2=8x\dots \left(1\right)$
Equation of chord of contact of $R(2,5)$ with respect to the parabola $y^2=8x$ is
$T=0\Rightarrow 5y=4(x+2)\Rightarrow 4x-5y+8=0\dots \left(2\right)$
Solving $\left(1\right)$ and $\left(2\right),$ we get
$P\equiv (8,8),Q\equiv (\frac{1}{2},2)$

Now, equation of circle having $PQ$ as a diameter is
$(x-8)(x-\frac{1}{2})+(y-8)(y-2)=0$
$\Rightarrow x^2+y^2-\frac{17x}{2}-10y+20=0$
Length of $y-$intercept
$=2\sqrt{f^2-c}$
$=2\sqrt{(-5{)}^2-20}$
$=2\sqrt{5}$