Let the focal chord of the parabola $P: y^{2}=4 x$ along the line $\mathrm{L}: \mathrm{y}=\mathrm{mx}+\mathrm{c}, \mathrm{m}>0$ meet the parabola at the points M and N . Let the line L be a tangent to the hyperbola $\mathrm{H}: \mathrm{x}^{2}-\mathrm{y}^{2}=4$. If O is the vertex of P and F is the focus of H on the positive x -axis, then the area of the quadrilateral OMFN is :

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Parabola

Let the focal chord of the parabola

P : y^{2} = 4 x

along the line

L : y = mx + c, m > 0

meet the parabola at the points M and N . Let the line L be a tangent to the hyperbola

H : x^{2} - y^{2} = 4

. If O is the vertex of P and F is the focus of H on the positive x -axis, then the area of the quadrilateral OMFN is :

Answer: (B)

The parabola

y^{2} = 4 x

has focus at (1,0). Since L is a focal chord, it passes through (1,0), so

c = - m

.

The hyperbola

\frac{x ^{2}}{4} - \frac{y ^{2}}{4} = 1

has focus at

F (22, 0)

. For line

y = m x + c

to be tangent to this hyperbola,

c^{2} = 4 m^{2} - 4

. Substituting

c = - m

m^{2} = 4 m^{2} - 4

, giving

m = \frac{2}{3}

(since

m > 0

).

The line is

y = \frac{2 x}{3} - \frac{2}{3}

. Substituting into

y^{2} = 4 x

\frac{4 ( x - 1 ) ^{2}}{3} = 4 x

, which gives

x^{2} - 5 x + 1 = 0

. The corresponding y-values satisfy

y^{2} - 23 y - 4 = 0

.

The area of quadrilateral OMFN with O at origin, F at

(22, 0)

, and M, N having y-coordinates

y_{1}, y_{2}

is:

\frac{1}{2} ∣2 2 (y_{2} - y_{1}) ∣ = 2 (y_{2} - y_{1})^{2} = 2 12 + 16 = 214

JEE Main · 2022 · 29 Jul 2022 · Shift 1 · Single correct option