Write the coordinates of the point on the curve

Question:

Write the coordinates of the point on the curve $y^{2}=x$ where the tangent line makes an angle $\frac{\pi}{4}$ with $x$-axis.

Solution:

Given that the curve $y^{2}=x$ has a point where the tangent line makes an angle $\frac{\pi}{4}$ with $x$-axis.

$\therefore$ Slope of the tangent $\frac{\mathrm{dy}}{\mathrm{dx}}=\tan 45^{\circ}=1$

$\because$ the point lies on the curve.

$y^{2}=x$

$\Rightarrow 2 y \frac{d y}{d x}=1$

$\Rightarrow \frac{d y}{d x}=\frac{1}{2 y}$

$\Rightarrow \frac{1}{2 y}=1$

$\Rightarrow y=\frac{1}{2}$

So, $\mathrm{x}=\frac{1}{4}$

Hence, the required point is $\left(\frac{1}{4}, \frac{1}{2}\right)$

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