If the length of the latus rectum of an ellipse with its major axis along x-axis and center at the origin, be 8. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

see full answer

High-Paying Jobs That Even AI Can’t Replace — Through JEE/NEET

🎯 Hear from the experts why preparing for JEE/NEET today sets you up for future-proof, high-income careers tomorrow.

An Intiative by Sri Chaitanya

$(4 \sqrt{3}, 2 \sqrt{2})$

$(4 \sqrt{2}, 2 \sqrt{3})$

$(4 \sqrt{3}, 2 \sqrt{3})$

$(4 \sqrt{2}, 2 \sqrt{2})$

answer is D.

(Unlock A.I Detailed Solution for FREE)

JEE

NEET

Foundation JEE

Foundation NEET

CBSE

Detailed Solution

Let the equation of the ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

$∴ \frac{2 b^{2}}{a} = 8 \Rightarrow b^{2} = 4 a$ and $2 a e = 2 b \Rightarrow e = \frac{b}{a}$

Since, $e^{2} = 1 - \frac{b^{2}}{a^{2}} \Rightarrow e^{2} = 1 - e^{2} \Rightarrow e = \frac{1}{\sqrt{2}}$

Now, $\frac{1}{2} = 1 - \frac{4 a}{a^{2}} \Rightarrow a = 8$ and $b = a e = 8 \cdot \frac{1}{\sqrt{2}} = 4 \sqrt{2}$

Equation of ellipse is $\frac{x^{2}}{64} + \frac{y^{2}}{32} = 1$

The point only in option (d) i.e., $(4 \sqrt{3}, 2 \sqrt{2})$ lies on the ellipse.

Watch 3-min video & get full concept clarity

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test