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 $p,h$ and $k$

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

Answer 1

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

$(p-2x{)}^2=16x$

$\Rightarrow p^2-4px+4x^2=16x$

$\Rightarrow 4x^2+x(-4p-16)+p^2=0$

$\Rightarrow x_1+x_2=\frac{4p+16}{4}=p+4$

$\therefore h=\frac{x_1+x_2}{2}=\frac{p+4}{2}$

$\frac{y_1+y_2}{2}=\frac{p-2x_1+p-2x_2}{2}=p-(x_1+x_2)$

$\therefore k=p-p-4=-4$

So, we have to check between options C and D.

Option C: $p=-2,h=2\Rightarrow x_1+x_2=2h=4$

$\Rightarrow k=p-(x_1+x_2)=-2-4=-6\ne -4$

Hence, option C is not correct.

Option D: $p=2,h=3\Rightarrow x_1+x_2=2h=6$

$\Rightarrow k=p-(x_1+x_2)=2-6=-4$

Hence, option D is correct.