Question

The point on the curve $y^2=x$ where the tangent makes an angle of $\frac{\pi }{4}$ with $X$-axis is

• A. $\left(\frac{1}{2},\frac{1}{4}\right)$
• B. $\left(\frac{1}{4},\frac{1}{2}\right)$
• C. $(4,2)$
• D. $(1,1)$

Answer 1

The correct option is B $\left(\frac{1}{4},\frac{1}{2}\right)$

The slope of the line whichh makes an angle of $\frac{\pi }{4}$ with the positive $X$-axis = $\tan \frac{\pi }{4}=1$
Let any point on the curve be $(t^2,t)$
$2y{y}^{\prime }=1{|}_{(t^2,t)}$
$\Rightarrow y^{\prime }=\frac{1}{2t}=1$
$\Rightarrow t=\frac{1}{2}$

So, The point is $\left(\frac{1}{4},\frac{1}{2}\right)$