Question

Consider an elipse having its foci at $A\left(z_1\right)\text{and}B\left(z_2\right)$ in the argand plane. If the eccentricity of the ellipse is $e$ and it is known that origin is an interior point of the ellipse, then $e$ lies in the interval

• A. $(0,\frac{|z_1-z_2|}{|z_1|+|z_2|})$
• B. $(0,\frac{|z_1-z_2|}{|z_1^2|+|z_2{|}^2})$
• C. $(0,\frac{|z_1|}{|z_2|})$
• D. $(0,\frac{|z_2|}{|z_1|})$

Answer 1

The correct option is A $(0,\frac{|z_1-z_2|}{|z_1|+|z_2|})$

Let $P\left(z\right)$ be any point on the ellipse. Then equation of the ellipse is
$|z-z_1|+|z-z_2|=\frac{|z_1-z_2|}{e}\cdots \left(1\right)$
If we replace
$z$ by $z_1$ or $z_2$ . L.H.S. of equation $\left(1\right)$ becomes $|z_1-z_2|$. Thus for any interior point of the ellipse, we have
$|z-z_1|+|z-z_2|<\frac{|z_1-z_2|}{e}$
It is given that origin is an interior point of the ellipse
$|0-z_1|+|0-z_2|<\frac{|z_1-z_2|}{e}$
$\Rightarrow 0<e<\frac{|z_1-z_2|}{|z_1|+|z_2|}$