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. $(9,7)$

Answer 1

Answer: (B)

$(2,4)$

The equation of the tangent $y^2=8x$ at $P(2,4)$ is
$4y=4(x+2)$
$\Rightarrow x-y+2=\mathrm{0....}\left(i\right)$
Let $(x_1,y_1)$ be the mid-point of chord $QR$. Then, equation of $QR$ is
$y{y}_1-4(x+x_1)-5=y_1^2-8x_1-5$
$\Rightarrow 4x-y{y}_1-4x_1+y_1^2=\mathrm{0....}\left(ii\right)$
Clearly, Eqs. (i) and (ii) represent the same line
$\therefore \frac{4}{1}=\frac{-y_1}{-1}=\frac{-4x_1+y_1^2}{2}$
$y_1=4;8=-4x_1+y_1$
$y_1=4;x_1=2$