Question

If $2x+y=p$ is a chord to the parabola $y^2=16x$ whose midpoint is $(h,k)$, then which of the following is/are true?

• A. $k=4$
• B. $k=-4$
• C. $2h-4=p$
• D. $2p-4=h$

Answer 1

Answer: (B), (C)

The correct options are
B $k=-4$
C $2h-4=p$

Parabola : $y^2=16x$
Given chord equation is $2x+y=p$
Midpoint is $(h,k)$

The equation of chord with midpoint $(h,k)$ is
$T=S_{1}ky-8(x+h)=k^2-16h\Rightarrow \left(\frac{-k}{4}\right)y+2x=\left(\frac{8h-k^2}{4}\right)$

Comparing both the equation, we get
$k=-4$ and $\frac{8h-16}{4}=p$
$\Rightarrow 2h-4=p$