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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)