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Question
If \[y = \sqrt{a^2 - x^2}\] prove that \[y\frac{dy}{dx} + x = 0\] ?
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Solution
\[\text{ We have, y } = \sqrt{a^2 - x^2}\]
\[\text{ Squaring both sides we get }, \]
\[ \Rightarrow y^2 = a^2 - x^2 \]
\[\text{ Differentiating both sides w . r . t x, we get, } \]
\[ \Rightarrow 2y\frac{d y}{d x} = \frac{d}{dx}\left( a^2 - x^2 \right)\]
\[ \Rightarrow 2y\frac{d y}{d x} = 0 - 2x \]
\[ \Rightarrow y\frac{d y}{d x} = - x\]
\[ \Rightarrow y\frac{d y}{d x} + x = 0\]
\[ \text{ Hence proved }\]
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