Question

 The point on the curve ${y^2} = x$ where the tangent makes an angle of $\dfrac{\pi }{4}$ with x-axis is?
A. (4,2)
B. ($\dfrac{1}{4},\dfrac{1}{2}$)
C. (1,1)
D. ($\dfrac{1}{2},\dfrac{1}{4}$)

Answer 1

Hint- We differentiate the equation of the curve and find the slope by taking it as $\tan \theta $ and then we equate the value of $\dfrac{{dy}}{{dx}}$ found in both cases and derive the point on the curve where the tangent makes angle.

Complete step by step answer:

Given that, ${y^2} = x$
Now, differentiate on both sides with respect to x to find the value of $\dfrac{{dy}}{{dx}}$
$
  2y\dfrac{{dy}}{{dx}} = 1 \\
  \dfrac{{dy}}{{dx}} = \dfrac{1}{{2y}}.....(1) \\
 $
Since, tangent makes an angle of $\dfrac{\pi }{4}$ with x-axis and also $\dfrac{{dy}}{{dx}}$ is the slope of the tangent
As you know, the slope of line is represented by $\tan \theta $, we get
\[\dfrac{{dy}}{{dx}} = \tan \dfrac{\pi }{4} = \tan 45^\circ = 1\], where $\pi = 180^\circ $
Now, substituting the value of \[\dfrac{{dy}}{{dx}}\] in equation (1), we get
\[
  \dfrac{1}{{2y}} = 1 \\
  \therefore y = \dfrac{1}{2} \\
 \]
Now, putting the value of y in equation ${y^2} = x$
${\left( {\dfrac{1}{2}} \right)^2} = x$
$\therefore x = \dfrac{1}{4}$
Hence, the coordinates of the point on the curve is $(\dfrac{1}{4},\dfrac{1}{2})$
Hence, the correct option is B.
Note-While solving these types of questions always remember to plot these questions using diagrams. This will make solving such questions easier and differentiate the curve equation as we require a slope and that is the only method to find slope which is also represented as m=$\tan \theta $.