Hyperbola
10.0 Asymptotes
10.0 Asymptotes
An asymptote of any curve is a straight line which touches it in two points at infinity. In case of hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, the equations of two asymptotes are $$y = \pm \frac{b}{a}x{\text{ or }}\frac{x}{a} \pm \frac{y}{b} = 0$$
Proof: Let $y=mx+c$ be the asymptote of the hyperbola $$\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1...(1)$$. Substituting the value of $y$ in $(1)$, we get
$$\begin{equation} \begin{aligned} \frac{{{x^2}}}{{{a^2}}} - \frac{{{{(mx + c)}^2}}}{{{b^2}}} = 1 \\ ({a^2}{m^2} - {b^2}){x^2} + 2{a^2}mcx + {a^2}({b^2} + {c^2}) = 0...(2) \\\end{aligned} \end{equation} $$
If the line $y=mx+c$ is an asymptote to the given hyperbola, then it touches the hyperbola at infinity. So, both roots of equation $(2)$ must be infinite i.e., $$\begin{equation} \begin{aligned} {a^2}{m^2} - {b^2} = 0{\text{ and }} - 2{a^2}mc = 0 \\ m = \pm \frac{b}{a}{\text{ and }}c = 0 \\\end{aligned} \end{equation} $$
Substituting the value of $m$ and $c$ in $y=mx+c$, we get $$\begin{equation} \begin{aligned} y = \pm \frac{b}{a}x \\ \frac{x}{a} \pm \frac{y}{b} = 0 \\\end{aligned} \end{equation} $$
