JEE Main 2022 · 29 Jul · Shift 1 · Mathematics Q17

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MathematicsSingle correct+4 / -1Question 17 of 90

Let the focal chord of the parabola P : $y^2=4x$ 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 :

2\sqrt6

2\sqrt{14}

4\sqrt6

4\sqrt{14}

Answer and worked solution

Official answer: B

Line. Through the focus (1, 0): $c=-m$ . Tangent to $\frac{x^2}4-\frac{y^2}4=1$ : $c^2=4m^2-4$ , so $m^2=\frac43$ .

Chord. $y^2-\frac4my-4=0$ gives $|y_M-y_N|=4\sqrt{\frac1{m^2}+1}=2\sqrt7$ . F = $(2\sqrt2,0)$ .

Area $\frac12\cdot2\sqrt2\cdot2\sqrt7=2\sqrt{14}$ .

(B)

Topics:Tangents and normals to conics Parabola Hyperbola

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JEE Main 2022 · 29 Jul · Shift 1, Mathematics question 17