For any general hyperbola
S:ax$^2$+2hxy+by$^2$+2gx+2fy+c=0
Given discriminant $\Delta\neq$ 0
How would I obtain the equation of its conjugate?
When I tried this I began to question what exactly the hyperbola and its conjugate had in common or differed in and the only possible approach I got was somehow finding the transverse and conjugate axes and interchanging them to give the conjugate but it was way too tedious.
However the final expression is
S':ax$^2$+2hxy+by$^2$+2gx+2fy+c-$\frac{2\Delta}{ab-h^2} $=0
This seems fairly similar to the equation of the assymptotes
L:ax$^2$+2hxy+by$^2$+2gx+2fy+c-$\frac{\Delta}{ab-h^2} $=0
What's the best way I can proceed with?