Ellipse 1 Question 1
1. An ellipse, with foci at $(0,2)$ and $(0,-2)$ and minor axis of length 4, passes through which of the following points?
(2019 Main, 12 April II)
(a) $(\sqrt{2}, 2)$
(b) $(2, \sqrt{2})$
(c) $(2,2 \sqrt{2})$
(d) $(1,2 \sqrt{2})$
Show Answer
Answer:
Correct Answer: 1. (a)
Solution:
- Let the equation of ellipse be
$$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $$
Since, foci are at $(0,2)$ and $(0,-2)$, major axis is along the $Y$-axis.
$$ \text { So, } \quad b e=2 $$
[where $e$ is the eccentricity of ellipse] and $2 a=$ length of minor axis $=4$
[given]
$$ \begin{array}{ll} \Rightarrow & a=2 \\ \because & e^{2}=1-\frac{a^{2}}{b^{2}} \\ \therefore & \frac{2}{b}^{2}=1-\frac{4}{b^{2}} \\ \Rightarrow & \frac{8}{b^{2}}=1 \Rightarrow b^{2}=8 \end{array} $$
Thus, equation of required ellipse is $\frac{x^{2}}{4}+\frac{y^{2}}{8}=1$
$\because e=\frac{2}{b}$ Now, from the option the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{8}=1$ passes through the point $(\sqrt{2}, 2)$.
