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Equation of a Line Passing through 2 Points

Consider an e...

Question

Consider an ellipse having its foci at $a (z_{1})$ and $B (z_{2})$ in the Argand plane. If the eccentricity of the ellipse be e and it is known that origin is an interior point of the ellipse, then prove that $e \in (0, \frac{| z_{1} - z_{2} |}{| z_{1} | + | z_{2} |})$ .

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Solution

Let $P (z)$ be any point on the ellipse. Then equation of the ellipse is
$| z - z_{1} | + | z - z_{2} | = \frac{| z_{1} - z_{2} |}{e}$ (1)
If we replace $z$ by $z_{1}$ or $z_{2}$ , L.H.S. of (1) 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
$| 0 - z_{1} | + | 0 - z_{2} | < \frac{| z_{1} - z_{2} |}{e}$
$⟹ e \in (0, \frac{| z_{1} - z_{2} |}{| z_{1} | + | z_{2} |})$
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