Find closest point on HoughLine

Given a hough line as {rho: number, theta: number} and a coordinate as {x: number, y: number}, how would I find the closest point on that hough line to the given coordinate?

So in the following sample graphic, I have the blue line and the red dot, and I'm looking for the white cross.

I'm using Typescript, but I'd appreciate answers in any language or format, since this is technically more of a math problem.

White cross is projection of red point C onto the line.

At first get components of normal vector to line

nx = cos(theta)
ny = sin(theta)

and direction vector of line is

dx = -ny
dy = nx

To find this projection, we need some point A at the line. Let's this point is

ax = rho * nx
ay = rho * ny

and get vector AC

acx = cx - ax
acy = cy - ay

Now calculate projection using dot product

coeff = dx*acx + dy*acy
px = ax + dx * coeff
py = ay + dy * coeff

Quick-made Python test (not well tested)

import math
def projection_onto_houghline(rho, theta, cx, cy):
   dx = -math.sin(theta)
   dy = math.cos(theta)
   ax = rho * dy
   ay = -rho * dx
   acx = cx - ax
   acy = cy - ay
   coeff = dx*acx + dy*acy
   px = ax + dx * coeff
   py = ay + dy * coeff
   return(px, py)

print(projection_onto_houghline(math.sqrt(2), math.pi/4, 0, 0))
print(projection_onto_houghline(math.sqrt(2), math.pi/4, 2.5, 2))

(1.0000000000000002, 1.0000000000000002)
(1.2500000000000002, 0.7500000000000002)