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**differentiate** This question came from my Calculus hw.

The circle $\displaystyle x^2 + y^2 + 2gx + 2fy + k = 0 $ intersects the rectangular hyperbola $\displaystyle x = ct $ $\displaystyle y= \frac{c}{t} $ in four points at $\displaystyle P_1 (ct_1, \frac{c}{t_1})$, $\displaystyle P_2(ct_2, \frac{c}{t_2}) $, $\displaystyle P_3(ct_3, \frac{c}{t_3}) $, $\displaystyle P_4(ct_4, \frac{c}{t_4}) $.

By showing that $\displaystyle t_1t_2t_3t_4 = 1$, Prove that the tangents at $\displaystyle P_1 $ and $\displaystyle P_2 $ meet on the diameter of the hyperbola perpendicular to $\displaystyle P_3P_4$

I can't show that $\displaystyle t_1t_2t_3t_4 = 1$. I think it has something to do with the angle in a semi-circle being 90 degrees or something. I can't do the next part either.