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

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

I can't show that $t_1t_2t_3t_4 = 1$. 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 $x^2 + y^2 + 2gx + 2fy + k = 0$ intersects the rectangular hyperbola $x = ct$ $y= \frac{c}{t}$ in four points at $P_1 (ct_1, \frac{c}{t_1})$, $P_2(ct_2, \frac{c}{t_2})$, $P_3(ct_3, \frac{c}{t_3})$, $P_4(ct_4, \frac{c}{t_4})$.
By showing that $t_1t_2t_3t_4 = 1$, Prove that the tangents at $P_1$ and $P_2$ meet on the diameter of the hyperbola perpendicular to $P_3P_4$
I can't show that $t_1t_2t_3t_4 = 1$. 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 $x = ct$ and $y= c/t$ 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.