If $\mathrm{P}(\mathrm{h}, \mathrm{k})$ be point on the parabola $\mathrm{x}=4 \mathrm{y}^{2}$, which is nearest to the point $\mathrm{Q}(0,33)$, then the distance of P from the directrix of the parabola $y^{2}=4(x+y)$ is equal to:

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Parabola

P (h, k)

be point on the parabola

x = 4 y^{2}

, which is nearest to the point

Q (0, 33)

, then the distance of P from the directrix of the parabola

y^{2} = 4 (x + y)

is equal to:

(A)2
(B)4
(C)8
(D)6✓ Correct

Answer: (D)

Solution

A point on

x = 4 y^{2}

, written

(4 k^{2}, k)

, closest to

Q (0, 33)

minimises

16 k^{4} + (k - 33)^{2}

; setting the derivative

64 k^{3} + 2 (k - 33) = 0

gives

32 k^{3} + k - 33 = 0

, whose only real root is

k = 1

, so

P = (4, 1)

. Rewriting the second parabola

y^{2} = 4 (x + y)

(y - 2)^{2} = 4 (x + 1)

shows a vertex at

(- 1, 2)

with

a = 1

, hence directrix

x = - 2

. The distance from

P (4, 1)

to this directrix is

4 - (- 2) = 6.

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Single correct option

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