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}}{{\varepsilon }_{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{4{{L}_{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 $ 2{{a}^{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 $