Question

If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint $(h,k)$, then which of the following is(are) possible value(s) of $p,h\text{and}k$ ?

• A. $p=-2,h=2,k=-4$
• B. $p=-1,h=1,k=-3$
• C. $p=2,h=3,k=-4$
• D. $p=5,h=4,k=-3$

Answer 1

The correct option is C $p=2,h=3,k=-4$

Parabola : $y^2=16x$
Chord equation : $2x+y=p$
Midpoint : $(h,k)$
Equation of chord with a given midpoint $(h,k)$: $T=S_1$
$ky-8(x+h)=k^2-16h\Rightarrow \left(\frac{-k}{4}\right)y+2x=\left(\frac{8h-k^2}{4}\right)$
Comparing both the equation,
$k=-4$ and $\frac{8h-16}{4}=p$
$\Rightarrow 2h-4=p$
Using the options given,
$p=2,h=3,k=-4$
Hence option C is correct.