Question

If $si{n}^{-1}$ a is the acute angle between the curves $x^2+y^2=4x$ and $x^2+y^2=8$ at (2, 2), then a= _____

• A. $1$
• B. $0$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{\sqrt{3}}{2}$

Answer 1

Answer: (C)

$\frac{1}{\sqrt{2}}$

Angle between the curves is given by the angle between their tangents at the given point.

For the curve $x^2+y^2-4x=0,2x+2y\frac{dy}{dx}-4=0$

$\Rightarrow 4\frac{dy}{dx}=4-4=0$

$\Rightarrow \frac{dy}{dx}=0$ at $(2,2)$

For the curve $x^2+y^2=8,2x+2y\frac{dy}{dx}=0$

$\Rightarrow \frac{dy}{dx}=\frac{-x}{y}=-1$ at $(2,2)$

$\tan a=\frac{0-(-1)}{1-0(-1)}=1$

$\Rightarrow \sin a=\frac{1}{\sqrt{1+1}}=\frac{1}{\sqrt{2}}$

$\therefore a={\sin }^{-1}\frac{1}{\sqrt{2}}$