$\left(y-y_1\right)=m\left(x-x_1\right)$

$x^2+y^2+2 g x+2 fy+c=0$

$\frac{y_1+f}{x_1+g} \times m=-1$

$m\left(y_1+f\right)+\left(x_1+g\right)=0$

$g=-x_1-m\left(y_1+f\right)$

$g^2+f^2-c$

$=(x_1+g)^2+(y_1+f)^2$

$c=g^2+f^2-\left(x_1+g\right)^2-(y_1+f)^2$

$x^2+y^2+2 x (-x_1-m(y_1+f))+2 f y+c=0$

$x^2+y^2+2 x\left(-x_1-m y_1\right)+f(2 y-2 m x) +g^2+f^2-\left(x_1+g\right)^2-\left(y_1 + f\right)^2 = 0$

$x^2+y^2+2 x\left(-x_1-m y_1\right)+f\left(2 y-2 m_x\right) -x_1^2-y_1^2-2g x_1-2fy_1=0 .$

substitute $g=-x_1-m\left(y_1+f\right)$

$x^2+y^2+2 x\left(-x_1-m y_1\right)+f(2 y-2 m x) -x_1^2-y_1^2+2 x_1\left(x_1+m\left(y_1+f\right))\right. -2 f y_1=0 .$

$(x+g)^2+(y-y_1)^2=\left(x_1+g\right)^2$

$x^2+2 g x+g^2 +\left(y-y_1\right)^2 =x_1^2+g^2+2 g_{x_1}$

$x^2+\left(y-y_1\right)^2+2 g\left(x-x_1\right)-x_1^2=0$

$\left(x-x_1\right)^2+2 x x_1-x_1^2+\left(y-y_1\right)^2+2 g\left(x-x_1\right)-x_1^2=0$

$\left(x-x_1\right)^2+\left(y-y_1\right)^2+2 g\left(x-x_1\right)+ 2xx_1 - 2x_1^2 = 0$

$\left(x-x_1\right)^2+\left(y-y_1\right)^2+\left(x-x_1\right)({2 g}+2 x_1)=0$

$\left(x-x_1\right)^2+\left(y-y_1\right)^2+k\left(x-x_1\right)=0$

$\left(x-x_1\right)^2+(y+f)^2=\left(y_1+f\right)^2$

$\left(x-x_1\right)^2+\left(y-y_1\right)^2+k\left(y-y_1\right)=0$

$\quad$

$x^2+y^2+2 x\left(-x_1-m y_1\right) +f(2 y-2 m x) -x_1^2-y_1^2+ 2 x_1\left(x_1+m\left(y_1+f\right)\right)-2 f y_1=0$

$\quad$

$(x-x_1)^2+(y-y_1)^2+K[(y-y_1)-m(x-x_1)] = 0$

General equation for family of circles: straight line is parallel to the x-axis

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

slope of chord

$=\frac{\left(y-y_1\right)}{\left(x-x_1\right)}$

$\therefore \quad \frac{\left(y-y_1\right)}{\left(x-x_1\right)} \frac{\left(y_1+f\right)}{\left(x_1+g\right)}=-1$

$(y-y_1)\left(y_1+f\right)+\left(x-x_1\right)\left(x_1+g\right)=0 .$

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

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

$P O^2=L^2+R^2$

$P O^2=\left(x_1+g\right)^2+(y_1+f)^2$

$L=\sqrt{P O^2-R^2}$

$=\sqrt{\left(x_1+g\right)^2+\left(y_1+f\right)^2-R^2}$

$L^2 = x_1^2+ 2 g x_1 + y_1^2 +2 f y_1 + g^2 +f^2 - R^2$

$S_1=\left(x-x_1\right)^2+\left(y-y_1\right)^2-L^2=0$

$S_2=x^2+y^2+2 g x+2 f y+c=0$

Equation for the chord of contact

OM = h $\triangle T_1 M O \sim \triangle T_1 P O$

$\frac{x}{L}=\frac{h}{R}=\frac{R}{\sqrt{R^2+L^2}}$

$x=\frac{R L}{\sqrt{R^2+L^2}},$ $h=\frac{R^2}{\sqrt{R^2+L^2}}$
$T_1 T_2=\frac{2 R L}{\sqrt{R^2+L^2}}$

$\operatorname{Area}\left(P T_1 O T_2\right)=L R$

$T_2$ Area $\left(O T_1 T_2\right)=h x$

$=\frac{R^3 L}{\left(R^2+L^2\right)}$

$\therefore$ Area $\left(\triangle P T_1 T_2\right)$

$=\text { Area }\left(P T_1 O T_2\right)-\text { Area } (OT_1T_2)$

$=L R-\left(R^3 L\right) /\left(R^2+L^2\right) =R L^3 /\left(R^2+L^2\right)$

$\angle T_1 P T_2 =2 \tan ^{-1}(R / L) \\ =\tan ^{-1}(R / L)+\tan ^{-1}(P/ L) \\ =\tan ^{-1}\left[\frac{(2 R / L)}{1-R^2 / L^2}\right]$

$S_1=x^2+y^2+2 g x+2f y+c =0$

$\left(x-x_1\right)^2+(y \left.-y_1\right)^2 =L^2$

$S_2=x^2+y^2-2 x x_1 -2 y y_1+x_1^2+y_1^2-L^2=0$

$S=S_1+\lambda S_2=0 \quad(\lambda \neq-1)$

$x^2+y^2+2 g x+2 f y+c+\lambda\left(x^2+y^2-2 x x_1\right.-2 y y_1+x_1^2+y_1^2 -L^2)=0$

$x_1^2+y_1^2+2 g x_1+2 fy_1+c +\lambda \left(x_1^2+y_1^2-2 x_1^2-2 y_1^2+x_1^2+y_1^2\right. \left.-L^2\right)=0$

$\Rightarrow \lambda =\frac{\left(x_1^2+y_1^2+2 g x_1+2 fy_1+c\right)}{L^2}$

$=1$