Answer to Question #293604 in Analytic Geometry for Elley

Question

Prove that the conic passing through the points of intersection of two rectangular hyperbolas is also a rectangular hyperbola.

Answer 1

Lets say S = 0 and S' = 0 are the two rectangular hyperbolas,

we then have a + b = 0, and a' + b' = 0.

Hence, in the conic

$S-\gamma S'=0$

The sum of the coefficients of x² and y²

$=(a-\gamma a')+(b-\gamma b')\\=(a+b)+\gamma (a'+b')=0$

Hence the conic

$S-\gamma S'=0$

itself a rectangular hyperbola.