JEE Advanced · 2017 · 21 May 2017 · Paper 1 · Parabola

If a chord, which is not a tangent, of the parabola

y^{2} = 16 x

has the equation

2 x + y = p

, and midpoint

(h, k)

, then which of the following is(are) possible value(s) of

p, h

and

k

Answer: (C)

Solution

For the parabola

y^{2} = 16 x

, the equation of a chord with midpoint

(h, k)

is given by

T = S_{1}

y k - 8 (x + h) = k^{2} - 16 h

, which simplifies to

y k - 8 x = k^{2} - 8 h

. Comparing with

2 x + y = p

(rewritten as

- 8 x + 4 y = 4 p

), we get

\frac{k}{4} = \frac{- 8}{- 8} = \frac{k ^{2} - 8 h}{4 p}

. This gives

k = 4

is incorrect; let's use

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

. So

k = - 4

and

k^{2} - 8 h = - 4 p

, giving

16 - 8 h = - 4 p

2 h - p = 4

. Checking option (C):

h = 3, p = 2

gives

2 (3) - 2 = 4

✓, and

k = - 4

✓.

JEE Advanced · 2017 · 21 May 2017 · Paper 1 · One or more correct options