JEE Advanced 2017 · Paper 1 · Mathematics Q2

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MathematicsMultiple correct+4 / -2Question 38 of 54

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 $k$ ?

p = -2, h = 2, k = -4

p = -1, h = 1, k = -3

p = 2, h = 3, k = -4

p = 5, h = 4, k = -3

Answer and worked solution

Official answer: C

Chord with midpoint $(h,k)$ . For $y^2=16x$ , $T=S_1$ gives
$ky-8(x+h)=k^2-16h,$
with slope $\dfrac8k$ .

Slope. It must equal $-2$ , so $k=-4$ .

Conditions. The chord $2x+y=p$ passes through $(h,-4)$ , so $p=2h-4$ . For a real chord (not a tangent) the midpoint must lie inside the parabola: $k^2<16h$ , i.e. $h>1$ .

Check the options.

(A) $p=2h-4=0\ne-2$ .
(B) and (D) have $k\ne-4$ .
(C) $p=2(3)-4=2$ , $k=-4$ , $h=3>1$ . ✓

(C)

Topics:Parabola

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JEE Advanced 2017 · Paper 1, Mathematics question 2