Question

The point $(x_1,y_1)$ lies inside the circle $x^2+y^2+2gx+2fy+c=0$, if $x_1^2+y_1^2+2g{x}_1+2f{g}_1+c$ is less than zero

• A. True
• B. False

Answer 1

The correct option is A

True

From the figure we can see that, PC$<r$. this is the condition for P to lie inside the circle.

The coordinate of P are $(x_1,y_1)$ and

$C\equiv (-g,-f)$

$r=\sqrt{g^2+f^2-c}$

PC$<r$.

$\Rightarrow \sqrt{(x_1+g{)}^2+(y_1+f{)}^2}<\sqrt{g^2+f^2-c}$

$\Rightarrow x_1^2+y_1^2+2g{x}_1+2f{g}_1+g_2+f_2<g_2+f_2-C$

$\Rightarrow x_1^2+y_1^2+2g{x}_1+2f{g}_1+c<0$

$\Rightarrow$ The given statement is correct