Question

An ellipse has eccentricity $\frac{1}{2}$ and one focus is at point $P\left(\frac{1}{2},1\right)$. A common tangent to the circle $x^2+y^2=1$ and hyperbola $x^2-y^2=1$ which is nearer to point $P$ is directrix of ellipse. Then co-ordinates of centre of ellipse are

• A. $\left(\frac{1}{3},\frac{1}{3}\right)$
• B. $(1,1)$
• C. $\left(\frac{1}{3},1\right)$
• D. $(1,\frac{1}{3})$

Answer 1

The correct option is C $\left(\frac{1}{3},1\right)$

Directrix is $x=1$

Disteance between directrix and focus is -
$\frac{a}{e}-ae=\frac{1}{2}$
$\Rightarrow 2a-\frac{a}{2}=\frac{1}{2}$
$\Rightarrow a=\frac{1}{3}$
$\alpha =\frac{1}{2}-ae=\frac{1}{2}-\frac{1}{6}=\frac{1}{3}$

$\Rightarrow$ centre $\equiv (\frac{1}{3},1)$