The tangent at any point on the curve $x^2 + y^2 = r^2$ meets the coordinate axes at $A$ and $B$. If lines are drawn through $A$ and $B$ parallel to the coordinate axes to intersect at $P$,find the locus of $P$.
- A$x^2 + y^2 = r^2$
- B$x^2 + y^2 = 4r^2$
- C$\frac{1}{x^2} + \frac{1}{y^2} = \frac{1}{r^2}$
- D$\frac{1}{x^2} - \frac{1}{y^2} = \frac{1}{r^2}$
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