Question

If foci are points $(0,1)(0,-1)$ and minor axis is of length $1$, then equation of ellipse is

• A. $\frac{x^2}{1/4}+\frac{y^2}{5/4}=1$
• B. $\frac{x^2}{5/4}+\frac{y^2}{1/4}=1$
• C. $\frac{x^2}{3/4}+\frac{y^2}{1/4}=1$
• D. $\frac{x^2}{1/4}+\frac{y^2}{3/4}=1$

Answer 1

Answer: (A)

$\frac{x^2}{1/4}+\frac{y^2}{5/4}=1$

Given that focii are $(0,1),(0,-1)$

Axis lies along $y-axis$

Distance between focii$\to 2be=2$

$\Rightarrow be=1$

Given that $2a=1$

$\Rightarrow a=\frac{1}{2}$

We know that $\Rightarrow e^2=1-\frac{a^2}{b^2}$

$\Rightarrow b^2{e}^2=b^2-a^2$

Substituting above obtained values in this expression we get,

$\Rightarrow 1=b^2-(\frac{1}{2}{)}^2$

$\Rightarrow 1=b^2-\frac{1}{4}$

$\Rightarrow b^2=\frac{5}{4}$

Thus equation of ellipse$\Rightarrow \frac{x^2}{\frac{1}{4}}+\frac{y^2}{\frac{5}{4}}=1$