# Thread: prove that tangents meet on the diameter

1. ## prove that tangents meet on the diameter

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.

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.
For the first part, substitute $\displaystyle x = ct$ and $\displaystyle y= c/t$ in the equation of the circle. This will give a quartic equation in t, in which the product of the four roots is the constant term.