A rectangular hyperbola and ellipse have coincident endpoints of latus rectum. Eccentricity of ellipse equals
Step-by-Step Solution
Key Concept: Rectangular hyperbola: $e_H=\sqrt2$. Latus rectum endpoint at $(ae_H,b^2/a)=(a\sqrt2,a^2/a)=(a\sqrt2,a)$. For ellipse $(x^2/A^2+y^2/B^2=1)$: latus rectum endpoint $(Ae_E,B^2/A)$. Set equal: $A e_E=a\sqrt2$ and $B^2/A=a$. Also $B^2=A^2(1-e_E^2)$: $A(1-e_E^2)=a$. Plus $Ae_E=a\sqrt2$: $e_E/(1-e_E^2)=\sqrt2\Rightarrow e_E=1-e_E^2\cdot...$ Solve: $e_E=1/\sqrt2$.
$e_E=1/\sqrt2$.
Correct Answer: (B) $\dfrac{1}{\sqrt2}$