# Orthogonal Circles

Calculus Level 5

$C_{1}: x^{2}+y^{2}=r^{2}, \quad C_{2}: (x-p)^{2}+(y-q)^{2}=r^{2}$

$$C_{1}$$ and $$C_{2}$$ defined above are two circles in 2D space with radius $$r\, (> 0)$$ and have $$n$$ points of intersection, $$(x_{i},y_{i})$$ for $$i \in \{1,\ldots,n\}$$.

If $$C_{1}$$ and $$C_{2}$$ are perpendicular at all points of intersection, which of the following statements are true?

I. $$x_{i}^{2}+y_{i}^{2}-qy_{i}-px_{i}=0$$.
II. As we move the center of $$C_{2}$$ along the curve $$q=a$$ for some constant $$a \neq 0$$, $$\frac{dr}{dp}=\frac{p}{r}$$.
III. As we move the center of $$C_{2}$$ along $$p+bq=0$$ for some constant $$b \neq 0$$, $$\frac{dr}{dq}=\frac{b^{2}q+q}{2r}$$.

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