Question:
The point on the curve $y^{2}=x$ where tangent makes $45^{\circ}$ angle 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. $(1,1)$
Solution:
Given that $y^{2}=x$
The tangent makes $45^{\circ}$ angle with $x$-axis.
So, slope of tangent $=\tan 45^{\circ}=1$
$\because$ the point lies on the curve
$\therefore$ Slope of the curve at that point must be 1
$2 y \frac{d y}{d x}=1$
$\Rightarrow \frac{d y}{d x}=\frac{1}{2 y}$
$\Rightarrow \frac{1}{2 y}=1$
$\Rightarrow y=\frac{1}{2}$
And $x=\frac{1}{4}$
So, the correct option is B