JEE Mathematics 2026 Question with Solution

Given: The ellipses $E_1$ and $E_2$ have the same eccentricity. For $E_1$, the distance between the foci is $8$, so if its focal distance from the center is $c$, then $2c=8$. Also, the lengths of their latus recta satisfy $2l_1^2=9l_2$.

Find: The distance between the foci of $E_2$.

From the solution, for $E_1$ the eccentricity is used as

Since the distance between the foci is $8$,

Hence,

Now use the ellipse relation

So,

The length of the latus rectum of $E_1$ is

Using the given relation

we get

For $E_2$, the solution uses the same eccentricity $e=\frac{4}{5}$ and writes

Solving these gives

Therefore, the distance between the foci of $E_2$ is $\dfrac{32}{5}$, so the correct option is D.

Given: Same eccentricity for both ellipses, $2l_1^2=9l_2$, and focal distance of $E_1$ is $8$.

Find: Focal distance of $E_2$.

1. For $E_1$, the extracted working gives

and

Therefore,

2. Then

3. Hence the latus rectum of $E_1$ is

4. Now apply the condition

Substituting $l_1=\frac{18}{5}$,

5. For $E_2$, the extracted working states

Using these, the final result obtained in the solution is

Thus, the required distance between the foci of $E_2$ is $\dfrac{32}{5}$.