Suppose we have a horizontal hyperbola with equation:[br][math]\frac{x^2}{a^2}-\frac{y^2}{b^2}=1[/math][br]What will the equation of the conjugate hyperbola be?
In the conjugate hyperbola, the orientation will be different, so we can switch [math]x[/math] and [math]y[/math]:[br][math]\frac{y^2}{a^2}-\frac{x^2}{b^2}=1[/math][br]But also, the roles of [math]a[/math] and [math]b[/math] have switched, since the [math]b[/math] value for the original hyperbola will now be taking us to the vertices [math]V_1'[/math] and [math]V_2'[/math]:[br][math]\frac{y^2}{b^2}-\frac{x^2}{a^2}=1[/math][br][br]In other words, the equation is basically the same but with the order we are subtracting the two fractions reversed.