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 midpoint of $QR$ is

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

Answer 1

Answer: (B)

$(2,4)$

Slope of tangent to parabola $y^2=8x$ at $P(2,4)$ is $1$.
Equation of tangent to the parabola at $P(2,4)$ is
$y-4=x-2$
or $x-y+2=0$ ......(i)
Let the middle point of chord $QR$ be $(h,k)$
Equation of chord of parabola $y^2=8x+5$ whose middle point is $(h,k)$ is $T=S_1$
i.e., $ky-4(x+h)-5=k^2-8h-5$
or $4x-ky+k^2-4h=0$ .....(ii)
Equations (i) and (ii) must be identical,
$\Rightarrow \frac{4}{1}=\frac{k}{1}=\frac{k^2-4h}{2}$
$\Rightarrow k=4$ and $8=k^2-4h$
$\Rightarrow h=2$
Hence, the required point is $(2,4)$.