Question

Equation of the ellipse whose eccentricity is $\frac{1}{2}$ and focii are $(\pm 1,0)$ will be:

• A. $\frac{x^2}{3}+\frac{y^2}{4}=1$
• B. $\frac{x^2}{4}+\frac{y^2}{3}=1$
• C. $\frac{x^2}{4}+\frac{y^2}{3}=\frac{4}{3}$
• D. None of these

Answer 1

The correct option is B $\frac{x^2}{4}+\frac{y^2}{3}=1$

Foci are on x-axis and distance between them = 2
$\therefore 2ae=2\Rightarrow a\left(\frac{1}{2}\right)=1\Rightarrow a=2$
Further
$b^2=a^2(1-e^2)\Rightarrow b^2=4(1-\frac{1}{4})$
$\Rightarrow b^2=3$
Obviously
Centre of the ellipse (mid-point of the focii) is the origin and its axes are
along coordinate axes. Hence its equation will be
$\Rightarrow \frac{x^2}{4}+\frac{y^2}{3}=1$