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NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Ellipse

NTA Abhyas 2020 · Mathematics · Ellipse

The focal chord of the parabola y^ 2 = 32x touches the ellipse x^ 2...

The focal chord of the parabola

y^{2} = 32x

touches the ellipse

\dfrac{x^{2}}{{4}^{2}} + \dfrac{y^{2}}{{2}^{2}} = 1

in the first quadrant at the point

Options

A $\left(2, \sqrt{3}\right)$
B $\left(\dfrac{\sqrt{3}}{2}, \dfrac{3}{2}\right)$
C $\left(\dfrac{4}{\sqrt{2}}, \dfrac{2}{\sqrt{2}}\right)$
D $\left(\dfrac{2}{\sqrt{2}}, \dfrac{4}{\sqrt{2}}\right)$

Correct answer

A. $\left(2, \sqrt{3}\right)$

Step-by-step solution

Let the tangent to the ellipse at

\left(4cos \theta, 2sin \theta\right)

~be

\dfrac{xcos \theta}{4} + \dfrac{ysin \theta}{2} = 1

∵

it passes through the focus~\left(8,0\right)

<br />

⇒ cos \theta = \dfrac{1}{2} ⇒ \theta = \dfrac{\pi}{3}

<br />

⇒

point is \left(\dfrac{4}{2}, \dfrac{2\sqrt{3}}{2}\right) ≡

\left(2, \sqrt{3}\right)

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