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A point inside a circle of radius $\sqrt{50}$ lies $2$ units directly below a point on the circle, and $6$ units directly to the right of a point on the circle. What is the distance from the center of the circle to this point?

Can't find anything, this was rated as an easy question so i think there will be just one concept for this. I tried power of a point and and coordinate geometry but I cant answer it

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Let the centre of the circle be at the origin, and let the point have coordinates $(p,q)$. Then $p^2+(q+2)^2=50 \ \text{(eq. 1)}$, and $(p-6)^2+q^2=50 \ \text{(eq. 2)}$, so $p^2+(q+2)^2=(p-6)^2+q^2 \ \text{(eq. 3)} $.

Expanding, we have:

$$p^2+q^2+4q+4=p^2-12p+36+q^2$$ $$4q+4=-12p+36$$ $$q+1=-3p+9$$ $$q=-3p+8 \ \text{(eq. 4)}$$

You can then substitute back into equation $1$ to find $p$. Then you can use equation $2$ and subtract $-12p+36$ from the LHS to find $p^2+q^2$, then $\sqrt{p^2+q^2}$.

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  • $\begingroup$ You might want to start off by saying, let the center of the circle be at the origin. $\endgroup$ – Gerry Myerson Nov 16 '18 at 11:51
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    $\begingroup$ @GerryMyerson Fixed. $\endgroup$ – Toby Mak Nov 16 '18 at 11:52

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