Circle

The circle touches two parallel lines p and q, and its center lies on line a, which is the secant of lines p and q.

Write the equation of the circle and determine the coordinates of the center and radius.

p: x-10 = 0
q: -x-19 = 0
a: 9x-4y+5 = 0

Correct result:

x_S =  -4.5
y_S =  -8.88
r =  NAN

Solution:

$$\begin{gathered} A=p\cap a=[10;0] \\ B=q\cap a=[-19;0] \\ S={\displaystyle \frac{AB}{2}}=[-4.5;-8.88] \\ t\perp p;t\perp q;S\in t \\ t:+y+1=0 \\ T=p\cap t=[10;NAN] \end{gathered}$$

$y_S=(23.75+(-41.5))\mathrm{/}2=-{\displaystyle \frac{71}{8}}=-8.88$

$$\begin{gathered} r=\mathrm{\mid }SA\mathrm{\mid }=\sqrt{(x_S-x_T{)}^2+(y_S-y_T{)}^2}=NAN \\ (x+4.5{)}^2+(x+8.88{)}^2=NA{N}^2 \end{gathered}$$

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