Coordinate Geometry
Area relation between ellipse and circles; eccentricity
MJMT_Full_Test_06
Grade 12
Question:
If area bounded by an ellipse with its auxiliary circle equals area bounded by its auxiliary circle and director circle, then eccentricity of ellipse is
$\sqrt{2\sin18°}$
$\sqrt{2\cos18°}$
$\sqrt{2\sin36°}$
$\sqrt{2\cos36°}$
Step-by-Step Solution
Key Concept: Auxiliary circle area $=\pi a^2$. Ellipse area$=\pi ab$. Director circle area$=\pi(a^2+b^2)/1=2\pi a^2$ (radius$=\sqrt{a^2+b^2}$). Condition: $\pi a^2-\pi ab=\pi(a^2+b^2)-\pi a^2\Rightarrow a^2-ab=b^2\Rightarrow(b/a)^2+(b/a)-1=0\Rightarrow b/a=(\sqrt5-1)/2=2\sin18°$. $e^2=1-b^2/a^2=1-2\sin18°+...$
$e=\sqrt{2\sin18°}$.
Correct Answer: 1