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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