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NDA 2010
Exam
NDA
Year
2010
Subject
Mathematics
Chapter
Ellipse

NDA 2010 · Mathematics · Ellipse

If the latus rectum of an ellipse is equal to half its minor axis,...

If the latus rectum of an ellipse is equal to half its minor axis, then what is its eccentricity?

Options

  1. A12\frac{1}{2}
  2. B3\sqrt{3}
  3. C32\frac{\sqrt{3}}{2}
  4. D12\frac{1}{\sqrt{2}}

Correct answer

C. 32\frac{\sqrt{3}}{2}

Step-by-step solution

\begin{aligned} & \text{ Latus rectum }=\frac{2 b^2}{a} \\ & \text{minor axis }=2 b \\ & \begin{aligned} \therefore \quad \frac{2 b^2}{a} & =\frac{1}{2} \times 2 b \\ \Rightarrow \quad b & =\frac{2 b^2}{a} \\ \Rightarrow \quad a & =2 b\end{aligned}\end{aligned}NowfromformulaNow from formulab^2=a^2\left(1-e^2\right)b2=4b2(1−e2)14=1−e2⇒e2=1−14e2=34⇒e=32\begin{aligned} b^2 & =4 b^2\left(1-e^2\right) \\ \frac{1}{4} & =1-e^2 \\ \Rightarrow \quad e^2 & =1-\frac{1}{4} \\ e^2 & =\frac{3}{4} \\ \Rightarrow \quad e & =\frac{\sqrt{3}}{2} \end{aligned}$
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