Line:
y = 5*x - 13
Circle:
(x-2)^2 + (y+3)^2 = 26
If we want the intersection points, we substitute the line y in the Circle equation:
(x-2)^2 + ((5*x - 13)+3)^2 = 26
(x-2)^2 + (5*x - 10)^2 = 26
(x-2)^2 + (5*(x - 2))^2 = 26
(x-2)^2 + 25*(x - 2)^2 = 26
26*(x-2)^2 = 26
(x-2)^2 = 1
x^2 - 4*x + 3 = 0
x_1 = 1
x_2 = 3
y_1 = 5*x_1 - 13 = -8 -> A = (1,-8)
y_2 = 5*x_2 - 13 = 2 -> B = (3,2)
Finding M:
If M is the midpoint of AB,
vec M = (vec A + vec B)/2
vec M = [1+3, -8+2]*1/2 = [2,-3]
M = (2,-3)
Perpendicular line y = m*x+c :
As shown in
https://www.mathsisfun.com/algebra/line-parallel-perpendicular.html,
m = -0.2
and (P_0 = M)
(y-y_0) = m*(x-x_0)
(y-(-3)) = -0.2*(x-2)
y+3 = -0.2*x + 0.4
y = -2.6 - 0.2*x is the equation of the perpendicular line to AB that crosses it in M.
Hope it helps
