There's a family of lines and where they intersect, the envelope, is a hyperbola. The hyperbola's center is at the origin and the $y$-axis is its axis of symmetry, and I know both $a$ and $b$. I'm also given an arbitrary point $(x_1, y_1)$ that lies on one of the lines but does NOT lie on the hyperbola. What I have to find is the point (points?) $(x_2, y_2)$ that lies on that same line, but also DOES lie on the
hyperbola. In other words, the point at which the line through $(x_1, y_1)$ is tangent to the hyperbola. How?
I've searched, without luck, for an equation that takes a point on such a line and the hyperbola's equation, and gives the tangent point. I do have the equation for the line tangent to the hyperbola at $(x_2, y_2)$, which I thought I could work backwards to find the tangent point, but I've lost my way. All help is appreciated!
Best Answer
If $$ {x^2\over a^2}-{y^2\over b^2}=1 $$ is the equation of the hyperbola, then $$ {xx_1\over a^2}-{yy_1\over b^2}=1 $$ is the equation of the line through the tangency points, for the tangents issued from $(x_1,y_1)$ (see here for a proof).
The solutions of the system formed by these equations will then give the coordinates of the required tangency points.