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Calculating the coordinates on the circumference using the radius

Hello friends,

I'm new on this forum. I'd be very grateful if you could please advise on how to solve this problem: just the approach would be greatly appreciated please (Happy) - I've been stuck on this for days - I'd be grateful if you could please help me.

I need to find the coordinates on the circumference whereby the radius of the circle bisects a chord and touches that particular point. I've attached two images illustrating the situation.

I've solved the first three parts without any problems, but the last one is really bugging me. The question is actually one of the challenging ones on the exercise:

4. a) Show that the points P(2, -1) and Q(6, 7) lie on the circle whose equation is:

$\displaystyle x^2 + y^2 - 12x - 4y + 15$

(again, because I'm new, in case that equation doesn't appear, the equation is x^2 + y^2 - 12x - 4y + 15).

I've done this by subbing P and Q into that equation. PQ therefore form a chord.

b) Find the coordinates of the point M which is the midpoint of the chord PQ.

Midpoint is (4, 3).

c) Find the equation of the radius of the circle that passes through M.

The equation is $\displaystyle y = 5 - 1/2x$ (y = 5 - 1/2 x) or otherwise rearranged to $\displaystyle 2y = 10 - x$ (2y = 10 - x)

PM is equidistant to MQ. The radius from C to M is perpendicular to the chord PQ.

d) Find the coordinates of the point R where the radius intersects the circle.

Right - this is the one. I've tried to substitute the equation of the radius of forms, $\displaystyle 2y = 10 - x$ and $\displaystyle y = 5 - 1/2x$ into

$\displaystyle x^2 + y^2 - 12x - 4y + 15$ but I'm not getting the correct answer.

I've tried other approaches like producing a right angle triangle and finding the distance between CM and then from MR, but I don't see how this helps even if I were to use the equation to calculate the distance between C and R, which I've tried with little luck.

I've drawn a diagram and it clearly shows that the coordinates of R are *approximately* (1.5, 4.2); the actual solution is (6-2sqrt5, 2+sqrt5).

I've looked at other websites and they all suggest that we treat these as simultaneous equations and substitute to find first x and then use the solution of x to find y.

I'd be very grateful if you could please advise if I'm doing anything incorrectly and what I need to do to get the correct solutions please.

Thank you once again for your kind attention.