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 sum of the coordinates of the mid point of $QR$ is ​​

Answer 1

Equation of tangent at point $P(2,4)$ to parabola $y^2=8x$ is
$4y=4(x+2)\Rightarrow 4x-4y+8=0\cdots \left(1\right)$

Let $(x_1,y_1)$ be the mid point of the chord $QR$
Equaion of chord of the parabola $y^2=8x+5$ whose mid point is $(x_1,y_1)$ will be
$T=S_1\Rightarrow y{y}_1-4(x+x_1)-5=y_1^2-8x_1-5\Rightarrow 4x-y{y}_1+y_1^2-4x_1=0\cdots \left(2\right)$

$\because$ Both equation $\left(1\right)$ and $\left(2\right)$ represents the same equation
Hence, comparing $\left(1\right)$ and $\left(2\right)$, we get
$\frac{4}{4}=\frac{y_1}{4}=\frac{y_1^2-4x_1}{8}$
$\Rightarrow y_1=4$
$\Rightarrow x_1=2$

$\therefore$The required sum is $4+2=6$