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Question
The equation to the normal to the curve y = sin x at (0, 0) is ___________ .
Options
x = 0
y = 0
x + y = 0
x − y = 0
Solution
x + y = 0
\[\text { Given }: \]
\[y = \sin x\]
\[\text { On differentiating both sides w.r.t.x, we get }\]
\[\frac{dy}{dx} = \cos x\]
\[\text { Slope of the tangent }= \left( \frac{dy}{dx} \right)_\left( 0, 0 \right) = cos 0 =1\]
\[\text { Slope of the normal }, m=\frac{- 1}{1}=-1\]
\[\text { Given }: \]
\[\left( x_1 , y_1 \right) = \left( 0, 0 \right)\]
\[ \therefore \text { Equation of the normal }\]
\[ = y - y_1 = m\left( x - x_1 \right)\]
\[ \Rightarrow y - 0 = - 1\left( x - 0 \right)\]
\[ \Rightarrow y = - x\]
\[ \Rightarrow x + y = 0\]
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