EllipseeasyFree

Eccentricity from Triangle Area Ratio in Ellipse | JEE

JEE Maths question with a full step-by-step solution.

Question
The area of the triangle inscribed in an ellipse bears the ratio 5:3\sqrt5:3 to the area of the triangle formed by joining points on the auxiliary circle corresponding to the vertices of the first triangle. Then the eccentricity of the ellipse is:
A23\dfrac{2}{3}correct
B34\dfrac{3}{4}
C53\dfrac{\sqrt5}{3}
D35\dfrac{3}{5}
Solution
Step 1: Relate the two areas. If P(θ),Q(β),R(α)P(\theta),Q(\beta),R(\alpha) are points on the ellipse, the corresponding auxiliary-circle points are obtained by replacing bsin⁡b\sin with asin⁡a\sin. Writing both areas as determinants,
Δellipse=12 ab∣cos⁡θsin⁡θ1cos⁡βsin⁡β1cos⁡αsin⁡α1∣,Δauxiliary=12 a2∣cos⁡θsin⁡θ1cos⁡βsin⁡β1cos⁡αsin⁡α1∣.\Delta_{\text{ellipse}}=\frac12\,ab\begin{vmatrix}\cos\theta&\sin\theta&1\\ \cos\beta&\sin\beta&1\\ \cos\alpha&\sin\alpha&1\end{vmatrix},\qquad \Delta_{\text{auxiliary}}=\frac12\,a^2\begin{vmatrix}\cos\theta&\sin\theta&1\\ \cos\beta&\sin\beta&1\\ \cos\alpha&\sin\alpha&1\end{vmatrix}.
 Step 2: Their ratio is
ΔellipseΔauxiliary=ba=53.\frac{\Delta_{\text{ellipse}}}{\Delta_{\text{auxiliary}}}=\frac{b}{a}=\frac{\sqrt5}{3}.
 Step 3: Then
e2=1−b2a2=1−59=49  ⇒  e=23.e^2=1-\frac{b^2}{a^2}=1-\frac59=\frac49 \;\Rightarrow\; e=\frac23.
 Correct answer: (1)
Still stuck on this question?Ask your doubt on WhatsApp
Similar questions
Ellipse · easy
The equation x210−a+y24−a=1\dfrac{x^2}{10-a}+\dfrac{y^2}{4-a}=1 represents an ellipse if:
Ellipse · easy
The eccentricity of the ellipse (x−3)2+(y−4)2=y29(x-3)^2+(y-4)^2=\dfrac{y^2}{9} is:
Ellipse · medium
An ellipse having foci at (3,3)(3,3) and (−4,4)(-4,4) and passing through the origin has eccentricity equal to:
Ellipse · medium
The foci of an ellipse are (−2,4)(-2,4) and (2,1)(2,1). The point (1,236)\left(1,\dfrac{23}{6}\right) is an extremity of the minor axis. What is the value of the eccentricity?

Solve more, learn faster

Sign up free to solve more JEE Maths questions and explore doMath — timed drills, mastery sprints, bookmarks, and chapter-wise progress tracking.

Sign up freeExplore doMath