JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Ellipse

The line

x = 8

is the directrix of the ellipse

E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

with the corresponding focus

(2, 0)

. If the tangent to E at the point P in the first quadrant passes through the point

(0, 43)

and intersects the x-axis at Q, then

(3 PQ)^{2}

is equal to ______.

Answer: 39

Solution

The directrix

x = \frac{a}{e} = 8

and focus

(a e, 0) = (2, 0)

give

a^{2} = \frac{a}{e} \cdot a e = 16

, so

a = 4

e = \frac{1}{2}

and

b^{2} = a^{2} (1 - e^{2}) = 12

. The tangent at

(x_{0}, y_{0})

\frac{x x _{0}}{16} + \frac{y y _{0}}{12} = 1

; passing through

(0, 43)

forces

\frac{4 3 y _{0}}{12} = 1

, so

y_{0} = 3

, and the ellipse then gives

x_{0}^{2} = 12

, i.e.

x_{0} = 23

in the first quadrant. Setting

y = 0

in the tangent gives

Q = (\frac{8}{3}, 0)

. Then

P Q^{2} = (23 - \frac{8}{3})^{2} + (3)^{2} = \frac{4}{3} + 3 = \frac{13}{3}

, so

(3 P Q)^{2} = 9 \cdot \frac{13}{3} = 39

JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Integer answer