Q38 / 91 · KVPY 2020 (SB/SX) · Mathematics

Let $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (b < a),$ be an ellipse with major axis $A B$ and minor axis $C D .$ Let $F_{1}$ and $F_{2}$ be its two foci, with $A, F_{1}, F_{2}, B$ in that order on the segment $A B .$ Suppose $∠ F_{1} C B = 90 ° .$ The eccentricity of the ellipse is

Solution

$\frac{b}{- a e} \times \frac{b}{a} = - 1$

$\Rightarrow b^{2} = a^{2} e$

$\Rightarrow a^{2} (1 - e^{2}) = a^{2} e$

$\Rightarrow e^{2} + e - 1 = 0$

$\Rightarrow e = \frac{- 1 + \sqrt{5}}{2}$

Hence, the correct answer is option: 'C'.