For some $\theta\in\left(0,\frac{\pi}{2}\right)$, if the eccentricity of the hyperbola, $x^2-y^2\sec^2\theta=10$ is $\sqrt{5}$ times the eccentricity of the ellipse, $x^2\sec^2\theta+y^2=5$, then the length of the latus rectum of the ellipse, is :

Option 3: $\frac{4\sqrt{5}}{3}$

JEE Main 2020 · MathsMultiple choiceSingle correctMediumMulti-step

JEE Main 2 September 2020, Shift 2, Maths Q66

Question 66 of 75 in this shift, Maths question 16 of 25.

For some

\theta\in\left(0,\frac{\pi}{2}\right)

, if the eccentricity of the hyperbola,

x^2-y^2\sec^2\theta=10

\sqrt{5}

times the eccentricity of the ellipse,

x^2\sec^2\theta+y^2=5

, then the length of the latus rectum of the ellipse, is :

(1) $2\sqrt{6}$
(2) $\sqrt{30}$
(3) $\frac{4\sqrt{5}}{3}$ Official answer
(4) $\frac{2\sqrt{5}}{3}$

Official answer

Option 3

NTA final key (Sep 2020).

Chapter: Conic Sections
Topic: Ellipse Eccentricity
Idea tested: Eccentricity calculation

Same idea in other shifts

Asked 11× in all

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30 Jun 2022, Shift 1 · Q15Let the eccentricity of the ellipse $x^2 + a^2y^2 = 25a^2$ be b times the eccentricity of the hyperbola $x^2 - a^2y^2 = 5$ , where a is the…HardSingle correct
25 Jul 2021, Shift 1 · Q61Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , $a^2>b^2$ , passes through $\left(\sqrt{\frac{3}{2}},1\right)$ and has eccentricity…HardSingle correct

Question text from the official JEE Main paper published by NTA; answer from the final answer key. Chapter, topic and idea tags, difficulty and skill are Lumi’s. Lumi analyses 108 of the 174 JEE Main shifts held from 2017 to 2026.