Q208 / 270 · NDA 2023 (Phase 1) · Mathematics
Paragraph:
Consider the following for the next two $(02)$ items that follow: $\mathrm{P}(x, y)$ is any point on the ellipse $x^2+4 y^2=1$. Let $E, F$ be the foci of the ellipse.
Question:
Consider the following points:
1. $\left(\frac{\sqrt{3}}{2}, 0\right)$
2. $\left(\frac{\sqrt{3}}{2}, \frac{1}{4}\right)$
3. $\left(\frac{\sqrt{3}}{2},-\frac{1}{4}\right)$
Which of the following points lie on latus rectum of ellipse?
Question: Consider the following points: 1. $\left(\frac{\sqrt{3}}{2}, 0\right)$ 2. $\left(\frac{\sqrt{3}}{2}, \frac{1}{4}\right)$ 3. $\left(\frac{\sqrt{3}}{2},-\frac{1}{4}\right)$ Which of the following points lie on latus rectum of ellipse?
- A1 and 2 only
- B2 and 3 only
- C1 and 3 only
- Correct answerD1,2 and 3
Solution
$b^2=a^2\left(1-e^2\right)$
$\frac{1}{4}=1-e^2 \Rightarrow e=\frac{\sqrt{3}}{2}$
focii $( \pm a e, 0)=\left( \pm \frac{\sqrt{3}}{2}, 0\right)$
Equation of latus rectum, $x= \pm \frac{\sqrt{3}}{2}$
$\therefore \quad$ All 3 points lie on latus rectum
Hence, the correct answer is option: 'D'.