Question

If the tangent at the point $P(2,4)$ to the parabola $y^2=8x$ meets the parabola $y^2=8x+5$ at $Q$ and $R$, then the mid-point of $QR$ is

• A. $(4,2)$
• B. $(2,4)$
• C. $(7,9)$
• D. none of these

Answer 1

The correct option is B $(2,4)$

The equation of tangent to the parabola $y^2=8x$ at $P(2,4)$ is
$4y=4(x+2)$
$\Rightarrow x-y+2=0\cdots \left(i\right)$


The equation of chord of parabola $y^2=8x+5$ whose middle point is $(h,k)$ is
$T=S_1$
$\Rightarrow ky-4(x+h)-5=k^2-8h-5$
$\Rightarrow 4x-ky+k^2-4h=0\cdots \left(ii\right)$
Equations (i) and (ii) must be identical. Therefore,
$\frac{4}{1}=\frac{k}{1}=\frac{k^2-4h}{2}$
$\Rightarrow k=4$
$8=k^2-4h\Rightarrow h=2$

Hence, the required point is $(2,4)$