Question

The differential equation of all ellipses with centres at the origin and the ends of one axis of symmetry is at (? 1,0), is

• A. $c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}$
• B. $I\propto \frac{1}{{\lambda }^4}$
• C. $\frac{{\lambda }_1}{{\lambda }_2}={\left[\frac{I_2}{I_1}\right]}^{\frac{1}{4}}={\left(\frac{4}{1}\right)}^{\frac{1}{4}}=\sqrt{2}:1$
• D. $K=\frac{M}{\sqrt{4L_2}}$

Answer 1

Answer: (C)

Let the ends of other axis of symmetry be $(0,\pm a)$ . Then, the equation of the ellipse is $x^2+\frac{y^2}{a^2}=1$ $\Rightarrow$ $a^2{x}^2+y^2=a^2$ ...(i) On differentiating both sides w.r.t. x, we get $2a^{2}x+2y\frac{dy}{dx}=0$ $\Rightarrow$ $a^2=-\frac{y}{x}.\frac{dy}{dx}$ ??? (ii) On eliminating $a^2$ from Eqs. (i) and (ii), we get $-y\frac{dy}{dx}+y^2=-\frac{y}{x}.\frac{dy}{dx}$ $\Rightarrow$ $(x^2-1)\frac{dy}{dx}-xy=0$