Question

The foci of an ellipse coincide with the vertices of a hyperbola and vertices of the ellipse coincide with the foci of the hyperbola. If $l_{1},l_{2}$ are their eccentricities then

• A. $l_{1}^{2} + l_{2}^{2} = 1$
• B. $l_{1}^{2} + l_{2}^{2}$=$l_{1}^{2}l_{2}^{2}$
• C. $l_{1} + l_{2} = 2$
• D. $l_{1}l_{2} = 1$

Answer 1

Answer: (A)

$l_{1}^{2} + l_{2}^{2} = 1$

Answer 2

Let a, b, l₁ are semi major axis, semi minor axis and

eccentricity of ellipse.

$\mathbf{a}^{\mathbf{1}}\mathbf{,}\mathbf{b}^{\mathbf{1}}\mathbf{,}\mathbf{l}_{\mathbf{2}}$ are semi transverse axis, semi conjugate axis and

eccentricity of Hyperbola.

Given (a$\mathcal{l}_{1},0)\mspace{6mu} =$ (a¹, 0) ⇒ $\frac{a}{a^{1}} = \frac{1}{\mathcal{l}_{1}}$

(a, 0) = $(a^{1}\mathcal{l}_{2},0)$ ⇒ $\frac{a}{a^{1}} = \mathcal{l}_{2}$

∴ $\mathcal{l}_{1}\mathcal{l}_{2} = 1$