Question

The angle between the parabolas $y^2=x$ and $x^2=y$ at the origin is:

• A. $2{\tan }^{-1}\left(\frac{3}{4}\right)$
• B. $2{\tan }^{-1}\left(\frac{4}{3}\right)$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (C)

$\frac{\pi }{2}$

The slope of the tangent to the parabola $y^2=x$ is given by
$2y.\frac{dy}{dx}=1$ or
$\frac{dy}{dx}=\frac{1}{2y}$
${\frac{dy}{dx}}_{y=0}=\infty$. Therefore, $tan\theta =\infty$, $\theta ={90}^0$ (...i)
The slope of the tangent to the parabola $x^2=y$ is given by
$\frac{dy}{dx}=2x$ or
${\frac{dy}{dx}}_{(x,y)=0}=0$. Therefore, $\varphi =0^0$ (...ii)
Hence the angle between the parabolas at the origin is
$\theta -\varphi$
$={90}^0-0^0$
$={90}^0$.