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

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Answer:

Correct Answer: 1. (a)

Solution:

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



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