Question

At what point of the curve y = x² does the tangent make an angle of 45° with the x-axis?

Answer 1

Let the required point be (x₁, y₁).
The tangent makes an angle of 45^o with the x-axis.
∴ Slope of the tangent = tan 45^o = 1

\[\text { Since, the point lies on the curve } . \]

\[\text { Hence }, {y_1}^2 = x_1 \]

\[\text { Now,} y^2 = x\]

\[ \Rightarrow 2y\frac{dy}{dx} = 1\]

\[ \Rightarrow \frac{dy}{dx} = \frac{1}{2y}\]

\[\text { Slope of the tangent } = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =\frac{1}{2 y_1}\]

\[\text { Given }:\]

\[\frac{1}{2 y_1} = 1\]

\[ \Rightarrow 2 y_1 = 1\]

\[ \Rightarrow y_1 = \frac{1}{2}\]

\[\text { Now,} \]

\[ x_1 = {y_1}^2 = \left( \frac{1}{2} \right)^2 = \frac{1}{4}\]

\[ \therefore \left( x_1 , y_1 \right) = \left( \frac{1}{4}, \frac{1}{2} \right)\]