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