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 > 0 y = m (x – 1) x24−y24=1 y=mx±4m2−4 ±4m2−4=−m, m>0 ⇒4m2−4=m M (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 > 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:

Option 1 - <math> <mrow> <mn>2</mn> <mroot> <mrow> <mn>6</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math>

Option 2 - <math> <mrow> <mn>2</mn> <mroot> <mrow> <mn>1</mn> <mn>4</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math>

Option 3 - <math> <mrow> <mn>4</mn> <mroot> <mrow> <mn>6</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math>

Option 4 - <math> <mrow> <mn>4</mn> <mroot> <mrow> <mn>1</mn> <mn>4</mn> </mrow> <mrow></mrow> </mroot> </mrow> </math>

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9 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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