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) of $p,h$ and $k$?

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

Answer 1

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

Parabola: $y^2=16x....................\left(1\right)(a=4)$

Equation of Chord: $2x+y=p......................\left(2\right)$

Midpoint of chord is $(h,k),$

Equation of chord when midpoint is given,

$T=S_1$

$=>ky-\frac{16}{2}(x+h)=k^2-16h$

$=>ky-8x+8h-k^2=0...................\left(3\right)$

Equation $\left(2\right)$ and $\left(3\right)$ represents the same, so comparing them,

$=>\frac{k}{1}=\frac{-8}{2}=\frac{8h-k^2}{-p}$

$=>k=-4.......................\left(c\right)$

$=>p=2h-4......................\left(d\right)$

Now by hit and trial method we have $k=-4$

$\left(c\right)$ and $\left(d\right)$,

For $\left(c\right)$, $-2\ne 2\left(2\right)-4$

For $\left(d\right)$, $2=2\left(3\right)-4$

$=>k=-4,p=2,h=3.$