PAIBOTLearn
Sign inSign up

Ellipses and hyperbolas

Conic sections

Japanese school year: Math C

What you learn

Learn definitions and standard equations for ellipses (constant sum of focal distances) and hyperbolas (constant absolute difference of focal distances), along with hyperbolic asymptotes. These curves are essential for describing celestial orbits and navigation systems. Prior knowledge of circle equations and geometric loci will help you understand these curves.

Go to practice

Key points

This is the standard equation of a horizontally elongated ellipse centered at the origin. Values aa and bb represent semi-major and semi-minor axes, and the sum of distances to the two foci is always 2a2a.

x2a2+y2b2=1(a>b>0)\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0)

This is the standard equation of a hyperbola opening left and right, centered at the origin. The difference in distance from any point on the curve to the two foci is always constant at 2a2a.

x2a2−y2b2=1(a>0,b>0)\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a > 0, b > 0)

These are the equations of the two asymptotes that the hyperbola approaches as it extends infinitely far from the origin. They serve as essential guides when sketching the hyperbola.

y=±baxy = \pm \frac{b}{a}x

Choose a set to practice.