An ellipse, with foci at (0,2) and (0,-2)
Question:

An ellipse, with foci at $(0,2)$ and $(0,-2)$ and minor axis of length 4, passes through which of the following points ?

1. (1) $(\sqrt{2}, 2)$

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

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

4. (4) $(1,2 \sqrt{2})$

Correct Option: 1

Solution:

Let the equation of ellipse :

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

Given that length of minor axis is 4 i.e. $a=4$. Also given be $=2$

$\because a^{2}=b^{2}\left(1-e^{2}\right) \Rightarrow 4=b^{2}-4 \Rightarrow b=2 \sqrt{2}$

Hence, equation of ellipse will be $\frac{x^{2}}{4}+\frac{y^{2}}{8}=1$

$\because(\sqrt{2}, 2)$ satisfies this equation.

$\therefore$ ellipse passes through $(\sqrt{2}, 2)$.