Hard conics question, need help ASAP!

P, Q are two variable points on the hyperbola xy = c², such that the tangent at Q passes through A, the foot of the ordinate of P. (i.e. A is the intersection o the y-axis and the horizontal line through P.) Show that the locus of T, the intersection of tangents at P and Q, is a hyperbola with the same asymptotes as the given hyperbola.

So confused... Please help?

Question

Hard conics question, need help ASAP!

P, Q are two variable points on the hyperbola xy = c², such that the tangent at Q passes through A, the foot of the ordinate of P. (i.e. A is the intersection o the y-axis and the horizontal line through P.) Show that the locus of T, the intersection of tangents at P and Q, is a hyperbola with the same asymptotes as the given hyperbola.

So confused... Please help?

mathteacher1729 · Answer

Pretty sure $ xy = c^2$ is not an equation of a hyperbola.  A hyperbola is of the form $$\frac{(x-h)^2}{a^2}\pm\frac{(y-k)^2}{b^2}=1$$

Where $(h,k)$ is the center and $a,b$ represent the lengths of the major & minor axes. (whichever is bigger is the "major").

anonymous · Answer

xy = c² is a rectangular hyperbola with branches in the first and third quadrants http://mathworld.wolfram.com/images/eps-gif/RectangularHyperbola_700.gif