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Hyperbola

The equation of a hyperbola centred at $$(x_0,y_0)$$, with transverse $$x$$-axis, and radii $$a\gt0$$ and $$b\gt0$$ is $$$ \frac{(x-x_0)^2}{a^2}-\frac{(y-y_0)^2}{b^2} = 1$$$ It passes through $$(x_0\pm a,y_0) $$ and has asymptotes $$$\displaystyle \frac{y}{b} = \pm\frac{x}{a}.$$$ The two branches intersect the transverse $$x$$-axis.

We divide the equation by $$x^2$$ and send $$x\to\infty$$ to find the asymptotes $$\displaystyle \lim_{x\to\infty}\frac{y^2}{x^2} = \frac{b^2}{a^2}.$$

The equation of a hyperbola with transverse $$y$$-axis is $$$ \frac{(y-y_0)^2}{b^2} -\frac{(x-x_0)^2}{a^2}= 1.$$$

Hyperbola with transverse $$y$$-axis (A) and hyperbola with transverse $$x$$-axis (B)
Hyperbola with transverse $$y$$-axis (A) and hyperbola with transverse $$x$$-axis (B)