Question

If \[y = \sqrt{a^2 - x^2}\] prove that  \[y\frac{dy}{dx} + x = 0\] ?

Answer 1

\[\text{ We have, y } = \sqrt{a^2 - x^2}\] 

\[\text{ Squaring both sides we get }, \]

\[ \Rightarrow y^2 = a^2 - x^2 \]

\[\text{ Differentiating both sides w . r . t x, we get, } \]

\[ \Rightarrow 2y\frac{d y}{d x} = \frac{d}{dx}\left( a^2 - x^2 \right)\]

\[ \Rightarrow 2y\frac{d y}{d x} = 0 - 2x \]

\[ \Rightarrow y\frac{d y}{d x} = - x\]

\[ \Rightarrow y\frac{d y}{d x} + x = 0\]

\[ \text{ Hence proved }\]