JEE Main 2022 · MathsMultiple choiceSingle correctHardMulti-step

JEE Main 30 June 2022, Shift 1, Maths Q15

Question 15 of 90 in this shift, Maths question 15 of 30, Section A.

Let 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 minimum distance between the curves

y = e^x

and

y = \log_e x

. Then

a^2 + \frac{1}{b^2}

is equal to :

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

Official answer

Option 4

NTA final key.

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

Same idea in other shifts

Asked 11× in all

5 Apr 2026, Shift 1 · Q9Let a focus of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be $S(4, 0)$ and its eccentricity be $\frac{4}{5}$ . If the point…EasySingle correct
1 Feb 2024, Shift 1 · Q12For $0<\theta<\pi/2$ , if the eccentricity of the hyperbola $x^2-y^2\mathrm{cosec}^2\theta=5$ is $\sqrt{7}$ times eccentricity of the…MediumSingle correct
1 Feb 2024, Shift 2 · Q13Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ . Let the line passing through $P$ and parallel to $y$ -axis meet the…HardSingle correct
6 Apr 2023, Shift 2 · Q2In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of…EasySingle correct
25 Jul 2022, Shift 2 · Q12If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the line $\frac{x}{7}+\frac{y}{2\sqrt{6}}=1$ on the $x$ -axis and the line…EasySingle correct
27 Jun 2022, Shift 1 · Q13Let the eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ , $a>b$ , be $\frac{1}{4}$ . If this ellipse passes through the point…EasySingle 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
9 Jan 2020, Shift 1 · Q66If $\mathrm{e}_1$ and $\mathrm{e}_2$ are the eccentricities of the ellipse, $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola,…EasySingle 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.