Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20257 AprilEvening Shift
Exam
JEE Main
Year
2025
Date
7 April
Shift
Evening Shift
Subject
Mathematics
Chapter
Ellipse

JEE Main 2025 · 7 April · Evening Shift · Mathematics · Ellipse

Let the length of a latus rectum of an ellipse...

Let the length of a latus rectum of an ellipse x2a2+y2b2=1\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 be 10 . If its eccentricity is the minimum value of the function f(t)=t2+t+1112f(\mathrm{t})=\mathrm{t}^2+\mathrm{t}+\frac{11}{12}, t∈R\mathrm{t} \in \mathbf{R}, then a2+b2\mathrm{a}^2+\mathrm{b}^2 is equal to :

Options

  1. A125
  2. B126
  3. C120
  4. D115

Correct answer

B. 126

Step-by-step solution

Length of LR =2b2a=10⇒5=\frac{2 b^2}{a}=10 \Rightarrow 5\mathrm{a}=\mathrm{b}2^2...(1)
f(t)=t2+t+1112df(t)dt=2t+1=0⇒t=−12 Min value of f(t)=(−12)2+(−12)+1112=14−12+1112=3−6+1112=812=23=ee2=1−b2a2⇒49=1−b2a2\begin{aligned} & \mathrm{f}(\mathrm{t})=\mathrm{t}^2+\mathrm{t}+\frac{11}{12} \\ & \frac{\mathrm{df}(\mathrm{t})}{\mathrm{dt}}=2 \mathrm{t}+1=0 \Rightarrow \mathrm{t}=\frac{-1}{2} \\ & \text{ Min value of } \mathrm{f}(\mathrm{t})=\left(\frac{-1}{2}\right)^2+\left(\frac{-1}{2}\right)+\frac{11}{12} \\ & =\frac{1}{4} \frac{-1}{2}+\frac{11}{12}=\frac{3-6+11}{12}=\frac{8}{12}=\frac{2}{3}=\mathrm{e} \\ & \mathrm{e}^2=\frac{1-\mathrm{b}^2}{\mathrm{a}^2} \Rightarrow \frac{4}{9}=\frac{1-\mathrm{b}^2}{\mathrm{a}^2}\end{aligned}
⇒b2a2=1−4a=5a⇒b2=5a2a\Rightarrow \frac{\mathrm{b}^2}{\mathrm{a}^2}=\frac{1-4}{\mathrm{a}}=\frac{5}{\mathrm{a}} \Rightarrow \mathrm{b}^2=\frac{5 \mathrm{a}^2}{\mathrm{a}}...(2)
\begin{aligned} & \text{ From (1) & (2) } \\ & 5 \mathrm{a}=\frac{5 \mathrm{a}^2}{\mathrm{a}} \Rightarrow \mathrm{a}=9, \quad \mathrm{~b}=\sqrt{45}=3 \sqrt{5} \\ & \therefore \mathrm{a} 2+\mathrm{b} 2=81+45=126\end{aligned}
Practice Ellipse on Quantrex Academy →

More from Ellipse

Consider the ellipse E given by (x^2)/(18) + (y^2)/(12) = 1 . Let H be the hyperbola… JEE Advanced 2026Passage: Consider the ellipses given by x^2 + 4y^2 = 1 and 4x^2 + y^2 = 1 . Question: Let… JEE Advanced 2026Let (x^2)/(f(a^2+7a+3)) + (y^2)/(f(3a+15)) = 1 represent an ellipse with major axis along… JEE Main 2026The eccentricity of an ellipse E with centre at the origin O is 3 2 and its directrices… JEE Main 2026Let x = 9 be a directrix of an ellipse E , whose centre is at the origin and eccentricity… JEE Main 2026Let a focus of the ellipse E: (x^2)/(a^2) + (y^2)/(b^2) = 1 be S(4, 0) and its… JEE Main 2026Let A be the point (3, 0) and circles with variable diameter AB touch the circle x^2 +… JEE Main 2026Let an ellipse (x^2)/(a^2) + (y^2)/(b^2) = 1 , a < b , pass through the point (4, 3) and… JEE Main 2026An ellipse has its center at (1,-2) , one focus at (3,-2) and one vertex at (5,-2) . Then… JEE Main 2026Let the ellipse E: x^ 2 144 + y^ 2 169 =1 and the hyperbola H: x^ 2 16 - y^ 2 λ^ 2 =-1… JEE Main 2026Let the length of the latus rectum of an ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1,(a>b) , be 30.… JEE Main 2026Let (h, k) lie on the circle C: x^ 2 +y^ 2 =4 and the point (2 h+1,3 k+2) lie on an… JEE Main 2026Let each of the two ellipses E_ 1 : x^ 2 a^ 2 + y^ 2 b^ 2 =1,(a>b) and E_ 2 : x^ 2 ~A^ 2… JEE Main 2026Let the line y-x=1 intersect the ellipse x^ 2 2 + y^ 2 1 =1 at the points A and B. Then… JEE Main 2026Let S and S^ be the foci of the ellipse x^ 2 25 + y^ 2 9 =1 and P(α, β) be a point on the… JEE Main 2026If the line α x+4 y= 7 , where α ∈ R , touches the ellipse 3 x^ 2 +4 y^ 2 =1 at the point… JEE Main 2026 Full Ellipse list Ellipse JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs