Question

The normal at the point (1, 1) on the curve 2y + x² = 3 is _____________ .

Options

• x + y = 0

• x − y = 0

• x + y + 1 = 0

• x − y = 1

Answer 1

`x − y = 0`

 

\[\text { Given }: \]

\[2y + x^2 = 3\]

\[ \Rightarrow 2\frac{dy}{dx} + 2x = 0\]

\[ \Rightarrow \frac{dy}{dx} = \frac{- 2x}{2} = - x\]

\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( 1, 1 \right) =-1\]

\[\text { Slope of the normal },m=\frac{- 1}{\text { Slope of the tangent }}=\frac{- 1}{- 1}=1\]

\[\text { Now }, \]

\[\left( x_1 , y_1 \right) = \left( 1, 1 \right)\]

\[ \therefore \text { Equation of the normal }\]

\[ = y - y_1 = m \left( x - x_1 \right)\]

\[ \Rightarrow y - 1 = 1 \left( x - 1 \right)\]

\[ \Rightarrow x - y = 0\]