P is a variable point on the circle $C: x^2 + y^2 - 12x + 8y + 20 = 0$. Q is the mid-point of OP where O is the origin. Find the equation of the locus of Q.
I'm not really too sure how to continue with the question, as all my attempts have too many variables per equation or are plain ridiculous. I understand that I need to provide other details but I don't really have any to provide.
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Since every point is mapped to another point half way from the origin, the resulting locus ($Q$) must be another circle with half the radius and with centre halfway from the origin. Essentially the point $Q$ functions as a pantograph with a scaling factor of $0.5$.
New centre: $$\frac 12 (6,8)=(3,4)$$
New radius: $$\frac 12 \sqrt{32}=\sqrt{8}$$
New equation: $$(x-3)^2+(y-4)^2=8$$
$$C: x^2+y^2-12x+8y+20=0$$ $$(x-6)^2+(y+4)^2=32$$
In polar form, let the point on the circle be $$P\equiv(4\sqrt2\cos\theta+6,4\sqrt2\sin\theta-4)$$
By mid-point formula, $$Q\equiv(2\sqrt2\cos\theta+3,2\sqrt2\sin\theta-2)$$
To get the locus of $Q$, $$(x-3)=2\sqrt2\cos\theta$$ $$(y+2)=2\sqrt2\sin\theta$$
Squaring and adding, $$(x-3)^2+(y+2)^2=8$$