If the area of the auxiliary circle of the el | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The area of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is given by $A_{e} = \pi ab$.The auxiliary circle of this ellipse is $x^{2

Vedclass · Accepted Answer

$\frac{\sqrt{3}}{2}$

Vedclass · Answer

(B) The area of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is given by $A_{e} = \pi ab$.The auxiliary circle of this ellipse is $x^{2} + y^{2} = a^{2}$,and its area is $A_{c} = \pi a^{2}$.According to the problem,$A_{c} = 2 A_{e}$.Substituting the values,we get $\pi a^{2} = 2 \pi ab$,which simplifies to $a = 2b$.The eccentricity $e$ of an ellipse is given by $e = \sqrt{1 - \frac{b^{2}}{a^{2}}}$.Substituting $a = 2b$,we get $e = \sqrt{1 - \frac{b^{2}}{(2b)^{2}}} = \sqrt{1 - \frac{b^{2}}{4b^{2}}} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$.

If the area of the auxiliary circle of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ $(a > b)$ is twice the area of the ellipse,then the eccentricity of the ellipse is

Similar Questions

$A$ thin metallic spherical shell contains a charge $Q$ on it. $A$ point charge $+q$ is placed at the centre of the shell and another charge $q'$ is placed outside it as shown in the figure. All the three charges are positive. The force on the charge at the centre is:

In which of the following is the $P - O - P$ bond present?

$A$ green light is incident from the water to the air-water interface at the critical angle $(\theta_c)$. Select the correct statement.

The $IUPAC$ name for the complex $[Co(NO_2)(NH_3)_5]Cl_2$ is:

The displacement of a particle varies according to the relation $x = 3 \sin(100t) + 8 \cos^2(50t)$. Which of the following is $INCORRECT$ about this motion?

Vedclass Test Series

Exam Paper Generator

Online Exam Module