Question

The equation of the tangent drawn to the circle $x^2+y^2=4$ at the point whose polar coordinates are given by $(2,\frac{\pi }{3})$ is

• A. $x+\sqrt{3}y=2$
• B. $\sqrt{3}x+y=2$
• C. $x+\sqrt{3}y=4$
• D. $\sqrt{3}x+y=4$

Answer 1

The correct option is C $x+\sqrt{3}y=4$

The given point is given in polar form, so
$(r,\theta )=(2,\frac{\pi }{3})x=r\cos \theta ,y=r\sin \theta \Rightarrow (x,y)=(2\cos \frac{\pi }{3},2\sin \frac{\pi }{3})\Rightarrow (x,y)=(1,\sqrt{3})$

Given circle is $x^2+y^2=4$
Now, the equation of the tangent is
$x{x}_1+y{y}_1-4=0$
$\Rightarrow x+\sqrt{3}y-4=0$