Question

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

Answer 1

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

Squaring both sides we get,

\[y^2 = x^2 + a^2 \]

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

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

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

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

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

\[\text{ Hence proved }\]