$\text { C: } x^2+y^2+2 g x+2 f y+c=0$
$P:(a, b)$

$O P \ =\sqrt{(-g-a)^2+(-f-b)^2} \\ =\sqrt{(g^2+a^2+2 a g+f^2+b^2+2fb }$

$O P < r \Rightarrow$ P lies inside the circle

$O P>r \Rightarrow$ P lies outside the circle

$O P=r \Rightarrow$ P lies on the circle

$OP

$\sqrt{g^2+a^2+2 a g+f^2+b^2+2 f b}<\sqrt{g^2+f^2-c}$

$a^2+b^2+2 a g+2 f b+c<0$

Example:

$C: x^2+y^2+6 x-8 y+4=0$

P :$(2,-1)$

O P =$\sqrt{(-5)^2+(5)^2}$

$=\sqrt{50}>\sqrt{21}$

C: $\quad x^2+y^2+2 g x+2 f y+c=0$

L: $\quad y=m x+d$

let- circle $C$ and st. line L intersect at- some point $P(a, b)$.

$a^2+b^2+2 g a+2 f b+c=0$

$b=m a+d$

$a^2+(m a+d)^2+2 g a+2 f(m a+d)+c=0$

$a^2\left(1+m^2\right)+2 a(\text { md }+m f+g) +\left(a^2+2 f d+c\right)=0$

Two distinct-real root $\left(a_0, a_1\right) ,\left(a_0, m a_0+d\right)$ $\Rightarrow\left(a_1, m a_1+d\right)$

Equal real roots $\left(a_0\right) \Rightarrow\left(a_0, m a_0+d\right)$

Both roots are complex $\rightarrow$ No point of intersection.

C: $x^2+y^2+2 x-4 y-4=0$

centre $(-1,2)$ and Radius = 3

$L: x+y=-5 \Rightarrow y=-(x+5)$

$P(x, y)$

$x^2+(x+5)^2 +2 x+4(x+5)-4=0$

$2 x^2+16 x+41=0$

$x=\frac{-16 \pm \sqrt{(256-328)}}{4}$

C: $x^2+y^2+2 g x+2 f y+c=0$

C: $x^2+y^2+2 g x+2 f y+c=0$

$a^2+2 g a+c=0$

$a_0 , a_1=-g \pm \sqrt{g^2-c}$

$g^2

• if $g^2=c$ then the circle “touches” the $x$-axis at the point $(-g,0)$.

if $g^2>c$,

$a_0, a_1$

$a_0=-g+\sqrt{g^2-c}$

$a_1=-g-\sqrt{g^2-c}$

the circle intersect the $x$-axis at two distinct points

$\left(-g+\sqrt{g^2-c}, 0\right),\left(-g-\sqrt{g^2-c}, 0\right)$

$\therefore$ intercept made by the circle on the $x$-axis $=2 \sqrt{g^2-c}$.

• if $f^2>c$ then the intersect made by the circle on the y-axis is $=2 \sqrt{f^2-c}.$

• if $f^2=c$ then the intersect made by the circle on the y -axis is $0.$

• if $f^2

If one of the diameters of the circle $x^2+y^2-2 x-6 y+6=0$ is a chord to some other circle centered at $(2,1)$ then the radius of the other circle is 3_.