JEE Main & Advanced Mathematics Conic Sections Asymptotes of a Hyperbola

An asymptote to a curve is a straight line, at a finite distance from the origin, to which the tangent to a curve tends as the point of contact goes to infinity.

The equations of two asymptotes of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] are \[{{x}^{2}}=4y\] or \[\frac{x}{a}\pm \frac{y}{b}=0\].

Some important points about asymptotes

(i) The combined equation of the asymptotes of the hyperbola \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=0\].

(ii) When \[b=a\] i.e. the asymptotes of rectangular hyperbola \[{{x}^{2}}-{{y}^{2}}={{a}^{2}}\] are \[y=\pm x\], which are at right angles.

(iii) A hyperbola and its conjugate hyperbola have the same asymptotes.

(iv) The equation of the pair of asymptotes differ the hyperbola and the conjugate hyperbola by the same constant only i.e., Hyperbola – Asymptotes = Asymptotes – Conjugated hyperbola or

\[\left( \frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}-1 \right)-\left( \frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}} \right)=\left( \frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}} \right)-\left( \frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}+1 \right)\].

(v) The asymptotes pass through the centre of the hyperbola.

(vi) The bisectors of the angles between the asymptotes are the coordinate axes.

(vii) The angle between the asymptotes of the hyperbola \[S=0\] i.e., \[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\] is \[2{{\tan }^{-1}}\frac{b}{a}\] or 2\[{{\sec }^{-1}}e\].

(viii) Asymptotes are equally inclined to the axes of the hyperbola.