7. Given $O(0,0)$ and $Q(1,2)$, consider the set of points $P$ for which $O P: P Q=5: 4$. It is known that this set is a circle. Find its center.
Answer: $\left(\frac{25}{9}, \frac{50}{9}\right)$
Solution: Suppose we have $P(x, y)$. This implies in particular that $4 O P=5 P Q$, or $16 O P^{2}=25 P Q^{2}$. Hence, we have
$$
16\left(x^{2}+y^{2}\right)=25(x-1)^{2}+25(y-2)^{2} \text {. }
$$

This simplifies to
$$
9 x^{2}-50 x+25+9 y^{2}-100 y+100=0
$$
and, completing squares,
$$
\left(3 x-\frac{25}{3}\right)^{2}+\left(3 y-\frac{50}{3}\right)^{2}=\left(\frac{25}{3}\right)^{2}+\left(\frac{50}{3}\right)^{2}-25-100=\frac{2000}{9} \text {. }
$$

Our center is then given by $\left(\frac{25}{9}, \frac{50}{9}\right)$.