Consider an ellipse having its foci 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 displaystyle e in left-0- frac-left-z-1-z-2right-left-z-1right- - left-z-2right-right-

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Toppr Admin · Accepted Answer

Let P-z- be any point on the ellipse- Then equation of the ellipse is-z-x2212-z1-z-x2212-z2-z1-x2212-z2-e -xA0- -xA0- -xA0- -xA0- -xA0- -xA0- -1-If-xA0-we replace z-xA0- by z1 or z2- L-H-S- of -1- becomes-xA0-z1-x2212-z2- Thus- for any interior point of the ellipse-xA0-we have-xA0-z-x2212-z1-z-x2212-z2-lt-z1-x2212-z2-eIt is given that origin is an interior point of the ellipse-0-x2212-z1-0-x2212-z2-lt-z1-x2212-z2-e-x27F9-e-x2208-0-z1-x2212-z2-z1-z2-Ans- 1

Consider an ellipse having its foci at a(z1) and B(z2) 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∈(0,|z1−z2||z1|+|z2|).

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 $e \in (0, \frac{| z_{1} - z_{2} |}{| z_{1} | + | z_{2} |})$ .