# Interconvertibility of formula from transverse hyperbola to conjugate hyperbola

While reading about Standard Hyperbola, I came across a statement regarding conjugate and transverse hyperbola :

Apart from the equation, all the formulas of conjugate hyperbolas, involve replacement of a by b and vice versa in equations for transverse hyperbola.

However this did not come out to be true when I tried testing it on my own. For example , let us consider the case for finding out the value of c in the equation of a line y = mx + c if it has to be a tangent.

For Standard Hyperbola x^2/a^2 - y^2/b^2 = 1 , the value of c comes out to be √a^2m^2 - b^2. On the other hand for standard Hyperbola x^2/a^2 - y^2/b^2 = -1 , the value of c comes out to be √b^2 - a^2m^2.

So, in case if I have to transform an equation meant for transverse hyperbolas , to obtain the corresponding formula for conjugate hyperbola , what should I do? (If I dont have to follow the main way of deriving a particular formula.. as most of them are lengthy to solve.)

• Substituting $a^2\to-a^2$, $b^2\to-b^2$ should work. Commented Jan 11 at 21:11
• But it does not work in the above formula which I have written Commented Jan 15 at 12:23
• Why not? $\sqrt{a^2m^2-b^2}\to\sqrt{-a^2m^2+b^2}$. Commented Jan 15 at 13:56
• Ohk got it , thanks :) Commented Jan 17 at 9:57