Question

What is the standard form of a hyperbola?

Answer 1

The standard form of a hyperbola

There are two types of standard equations of hyperbola.
Case I:

• When the transversal axis is $x$axis and the transverse axis is $y$ axis, then the standard equation of the hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

• The coordinates of the vertices are $\left(\pm a,0\right)$.
• The eccentricity of the hyperbola is $e=\sqrt{1+\frac{b^2}{a^2}}$.
• The equations of directrix are $x=\pm \frac{a}{e}$.
• The coordinates of the foci are $\left(\pm ae,0\right)$.
  
  
  
  Case II:
• When the transversal axis is $y$axis and the transverse axis is $x$ axis, then the standard equation of the hyperbola is $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$.

• The coordinates of the vertices are $\left(0,\pm a\right)$.
• The eccentricity of the hyperbola is $e=\sqrt{1+\frac{b^2}{a^2}}$.
• The equations of directrix are $y=\pm \frac{a}{e}$.
• The coordinates of the foci are $\left(0,\pm ae\right)$.