Question

State whether the given statement is True or False. The point $(1,2)$ lies inside the circle $x^2+y^2-2x+6y+1=0.$

Answer 1

We have point $(1,2)$ and circle
$x^2+y^2-2x+6y+1=0$.
Let $S_{=}{x}^2+y^2-2x+6y+1\dots \left(1\right)$
Substitute $x=1,y=2$ in equation $\left(1\right)$
$\Rightarrow S_1=1+4-2\times 1+6\times 2+1$
$\Rightarrow S_1=16>0$

Thus, the point lies outside the circle.

Hence, statement is false.