KCET · Maths · Ellipse
If the area of the auxillary circle of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(\) where, \(a>b)\) is twice the area of the ellipse, then the eccentricity of the ellipse is
- A \(\frac{1}{\sqrt{3}}\)
- B \(\frac{1}{2}\)
- C \(\frac{1}{\sqrt{2}}\)
- D \(\frac{\sqrt{3}}{2}\)
Answer & Solution
Correct Answer
(D) \(\frac{\sqrt{3}}{2}\)
Step-by-step Solution
Detailed explanation
Equation of auxiliary circle of the ellipse,
\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \text { is } x^{2}+y^{2}=a^{2}\)
\(\therefore\) Area of auxiliary circle \(=\pi a^{2}\)
and area of an ellipse \(=\pi a b\)
Now, according to the question
\(\begin{aligned}
\pi a^{2}=2(\pi a b) \\
\Rightarrow \quad a b &=2 b \Rightarrow b=\frac{a}{2}...(i)
\end{aligned}\)
\(\therefore\) Eccentricity of an ellipse
\(\begin{aligned}
&=\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}=\sqrt{\frac{a^{2}-\frac{a^{2}}{4}}{a^{2}}} \quad \text { [from Eq. (i)] } \\
&=\sqrt{\frac{3 a^{2}}{4 a^{2}}}=\frac{\sqrt{3}}{2}
\end{aligned}\)
\(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \text { is } x^{2}+y^{2}=a^{2}\)
\(\therefore\) Area of auxiliary circle \(=\pi a^{2}\)
and area of an ellipse \(=\pi a b\)
Now, according to the question
\(\begin{aligned}
\pi a^{2}=2(\pi a b) \\
\Rightarrow \quad a b &=2 b \Rightarrow b=\frac{a}{2}...(i)
\end{aligned}\)
\(\therefore\) Eccentricity of an ellipse
\(\begin{aligned}
&=\sqrt{\frac{a^{2}-b^{2}}{a^{2}}}=\sqrt{\frac{a^{2}-\frac{a^{2}}{4}}{a^{2}}} \quad \text { [from Eq. (i)] } \\
&=\sqrt{\frac{3 a^{2}}{4 a^{2}}}=\frac{\sqrt{3}}{2}
\end{aligned}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- In a group \((G, *)\), for some element \(a\) of \(G\), if \(a^{2}=e\), where \(e\) is the identity element, thenKCET 2013 Medium
- The value of \( \tan \frac{\pi}{8} \) is equal toKCET 2016 Easy
- If \( f(x)=\left\{\begin{aligned} K x^{3} ; & \text { if } x \leq 2 \\ 3 ; & \text { if } x>2 \end{aligned}\right. \) is continuous at \( x=2 \), then the value of \( K \) isKCET 2017 Easy
- Write the set builder form \( A=\{-1,1\}: \)KCET 2015 Medium
- The inverse of the matrix \( \left[\begin{array}{rrr}2 & 5 & 0 \\ 0 & 1 & 1 \\ -1 & 0 & 3\end{array}\right] \) isKCET 2019 Easy
- If \((\mathbf{a} \times \mathbf{b})^{2}+(\mathbf{a} \cdot \mathbf{b})^{2}=144\) and \(|\mathbf{a}|=4\), then \(|\mathbf{b}|\) is equal toKCET 2012 Easy
More PYQs from KCET
- An induced current of 2 A flows through a coil. The resistance of the coil is \(10 \Omega\). What is the change in magnetic flux associated with the coil in 1 ms ?KCET 2024 Easy
- Which of the following is not a group with respect to the given operation?KCET 2007 Easy
- Three polaroid sheets \(P_{1}, P_{2}\) and \(P_{3}\) are kept parallel to each other such that the angle between pass axes of \(P_{1}\) and \(P_{2}\) is \(45^{\circ}\) and that between \(P_{2}\) and \(P_{3}\) is \(45^{\circ}\). If unpolarised beam of light of intensity \(128 \mathrm{Wm}^{-2}\) is incident on \(P_{1}\). What is the intensity of light coming out of \(P_{3}\) ?KCET 2020 Easy
- The following four wires of length \(L\) and radius \(r\) are made of the same material. Which of these will have the largest extension, when the same tension is applied?KCET 2011 Medium
- The energy gap in case of which of the following is less than \( 3 \mathrm{eV} ? \)KCET 2017 Hard
- Primary endosperm nucleus is formed by fusion ofKCET 2023 Hard