Ellipse
Relation Between Semi-axes and Eccentricity
nta_pyq_2024_jan
Grade 11
Question:
If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:
$\dfrac{\sqrt{5}}{3}$
$\dfrac{\sqrt{3}}{2}$
$\dfrac{1}{\sqrt{3}}$
$\dfrac{2}{\sqrt{5}}$
Step-by-Step Solution
Key Concept: Minor axis length $=2b$, distance between foci $=2ae$. Condition: $2b=\frac{1}{2}(2ae)\Rightarrow2b=ae$. Use $b^2=a^2(1-e^2)$ to solve for $e$.
$2b=ae\Rightarrow\frac{b}{a}=\frac{e}{2}$. $b^2=a^2(1-e^2)\Rightarrow\frac{e^2}{4}=1-e^2\Rightarrow e=\frac{2}{\sqrt{5}}$.
Correct Answer: 4