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. $(2,4)$
• B. $(4,2)$
• C. $(7,9)$
• D. None

Answer 1

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

Given Parabola: $y^2=8x..............\left(1\right)$

$y^2=8x+5..............\left(2\right)$

Tangent at $(2,4)$ on $\left(1\right)$, $=>4y=4(x+2)$

$=>y=x+2............\left(3\right)$

Point of intersection of $\left(3\right)$ and $\left(2\right)$

$=>y^2=8(y-2)+5$

$=>y^2-8y+11=0$

$=>y=4\pm 2\sqrt{5}$

$=>x=2\pm 2\sqrt{5}$

Point of intersection: $Q(2+2\sqrt{5},4+2\sqrt{5})$and$R(2-2\sqrt{5},4-2\sqrt{5})$

Mid point of $Q$ and $R$: $(\frac{2+2\sqrt{5}+2-2\sqrt{5}}{2},\frac{4+2\sqrt{5}+4-2\sqrt{5}}{2})$

$:(2,4).$