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First find the midpoint and slope of points (h, k) and (3h, -5k)

Midpoint = (h+3h)/2, (6k+k)/2 = (2h, -2k)

Slope = (-5k-k)/(3h-h) = -6k/2h = -3k/h

Then find the perpendicular slope which is the positive reciprocal of -3k/h which is h/3k

Then substitute these values into the formula of y-y1 = m(x-x1) :-

y-(-2k) = h/3k(x -2h)

y = hx/3k -2h2/3k -2k

Multiply all terms by 3k and the perpendicular bisector equation can be expressed in the form of :-

hx -3ky -2h2 -6k2 = 0

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Q: How do you work out the equation for the perpendicular bisector of the straight line joining h k and 3h -5k showing details of your work?
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