Let the focal chord of the parabola P : y2 = 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 : x2 – y2 = 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:

L : y = mx + c, m > 0y = m (x – 1)x24−y24=1y=mx±4m2−4±4m2−4=−m, m>0⇒4m2−4=mM (5+212, 3+7), N (5−212, 3−7)=2 (27)=2

Let the focal chord of the parabola P : y2 = 4x along the line L : y = mx + c, m &gt; 0 meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola H : x2 – y2 = 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:

Ask & Answer Question

Option 1 -

$2 \sqrt[]{6}$

Option 2 -

$2 \sqrt[]{14}$

Option 3 -

$4 \sqrt[]{6}$

Option 4 -

$4 \sqrt[]{14}$

2 Views | Posted 2 months ago

Answered by

Vishal Baghel | Contributor-Level 10

2 months ago

Correct Option - 2

Detailed Solution:

L : y = mx + c, m > 0

y = m (x – 1)

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

$y = m x \pm \sqrt[]{4 m^{2} - 4}$

$\pm \sqrt[]{4 m^{2} - 4} = - m, m > 0$

$\Rightarrow \sqrt[]{4 m^{2} - 4} = m$

$M (\frac{5 + \sqrt[]{21}}{2}, \sqrt[]{3} + \sqrt[]{7}), N (\frac{5 - \sqrt[]{21}}{2}, \sqrt[]{3} - \sqrt[]{7})$

$=$ $2$ $(27)$ $=$ $2$

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