JEE Main · 2024 · 06 Apr 2024 · Shift 1 · Parabola

Let a conic

C

pass through the point

(4, - 2)

and

P (x, y)

x \geq 3

, be any point on

C

. Let the slope of the line touching the conic

C

only at a single point

P

be half the slope of the line joining the points

P

and

(3, - 5)

. If the focal distance of the point

(7, 1)

C

d

, then

12 d

equals ____.

Answer: 75

Solution

The tangent-slope condition gives

2 ln (y + 5) = ln (x - 3) + C

; passing through

(4, - 2)

fixes

C = 2 ln 3

, so

(y + 5)^{2} = 9 (x - 3)

- a parabola with

4 a = 9

a = \frac{9}{4}

, vertex

(3, - 5)

, opening along

+ x

.
Focal distance of

(7, 1)

d = (x - 3) + a = (7 - 3) + \frac{9}{4} = \frac{25}{4}

12 d = 12 \cdot \frac{25}{4} = 75.

JEE Main · 2024 · 06 Apr 2024 · Shift 1 · Integer answer

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