Hyperbola: locus and standard equation

A hyperbola is the locus of a point P that moves so that the difference of its distances from two fixed points (foci) is a constant, 2a, which is less than the distance between the foci.

Setting up

Take the foci F₁(−c, 0) and F₂(c, 0), with the origin at their mid-point and c > a > 0. Let P(x, y) be any point of the locus, so

|PF₁ − PF₂| = 2a

Derivation

Take PF₁ − PF₂ = 2a (the other branch gives the same final equation):

√((x+c)² + y²) − √((x−c)² + y²) = 2a

Move the second root to the right and square:

(x+c)² + y² = 4a² + 4a√((x−c)² + y²) + (x−c)² + y²

Expanding, the terms x², y², c² cancel and 4cx = 4a² + 4a√((x−c)² + y²), so

cx − a² = a√((x−c)² + y²)

Square again:

c²x² − 2a²cx + a⁴ = a²x² − 2a²cx + a²c² + a²y²

(c² − a²)x² − a²y² = a²(c² − a²)

Since c > a, put b² = c² − a² (b > 0):

b²x² − a²y² = a²b²

Divide by a²b²:

x²/a² − y²/b² = 1

This is the standard equation of the hyperbola.

Properties read from it

• Centre (0, 0); vertices (±a, 0); transverse axis 2a, conjugate axis 2b.
• Foci (±ae, 0), where b² = a²(e² − 1) and e = c/a > 1.
• Directrices x = ±a/e. Asymptotes y = ±(b/a)x.

If the foci are on the y-axis, the equation becomes y²/a² − x²/b² = 1.