Matrix Matching : Equation of circle $x^2+y^2-2x+10y-38=0$

A) A) Polar equation $S_1=0$

$\begin{array}{l}x{x}_1+y{y}_1-(x+x_1)+5(y+y_1)-38=0\\ 4x+3y-(x+4)+5(y+3)-38=0\\ 4x+3y-x-4+5y+15-38=0\\ 3x+8y=27\end{array}$

B) Equation of tangent $S_1=0$

$\begin{array}{l}x{x}_1+y{y}_1-(x+x_1)+5(y+y_1)-38=0\\ 9x-5y-(x+9)+5(y-5)-38=0\\ 9x-5y-x-9+5y-25-38=0\\ 8x=72\\ \therefore x=9\end{array}$

C) Equation of the tangent $S_1=0$

$\begin{array}{l}x{x}_1+y{y}_1-(x+x_1)+5(y+y_1)-38=0\\ -7x-5y-(x-7)+5(y-5)-38=0\\ -7x-5y-x+7+5y-25-38=0\\ -8x=56\\ x=-7\to \left(1\right)\end{array}$

Slope of equation (1) $=\frac{-1}{0}m$

$\begin{array}{l}{m}_1\times m_2=-1\\ \frac{-1}{0}\times m_2=-1\\ m_2=0\end{array}$

Equation of norma

l$\begin{array}{l}y-y_1=m(x-x_1)\\ y+5=0(x+7)\\ y+5=0\end{array}$

D) $\sqrt{S_{11}}=\sqrt{x_1^2+4_1^2-2x_1+10y_1-38}$

$\begin{array}{l}=\sqrt{0+16-0+40-38}\\ =\sqrt{56-38}\\ =\sqrt{18}\end{array}$