• question_answer At what point on the curve $y={{x}^{2}}$ does the tangent make an angle of $45{}^\circ$ with the X-axis?

Given equation of curve is Now,$y={{x}^{2}}$ Slope of the tangent $=\tan 45{}^\circ =1$ $y'=2x$ $\Rightarrow$$2x=1\Rightarrow x=\frac{1}{2}.$ On putting the value of x in Eq. (i), we get $y=\frac{1}{4}$Hence, the required point is $\left( \frac{1}{2},\,\,\frac{1}{4} \right)$.