Hyperbola and its equation

Refer to Maths is fun.

Hyperbola can be defined as a curve where the distances of any point from:

• A point (the focus)
• A (the directrix) are always in the same ratio.

Equation of Hyperbola

Equation of hyperbola opens on the X-axis:

Equation of hyperbola opens on the Y-axis:

Features:

• Two vertices:
• (±a, 0) when it opens on x-axis.
• (0, ±b) when it opens on y-axis.
• Two asymptotes:
• y = (b/a)x
• y = −(b/a)x

Foci of hyperbola

Any point P on the hyperbola to both focuses

|D₁ + D₂| = CONSTANT

How to find foci of hyperbola

f² = a² + b²

f is the focal length, which is the distance from the focus to the centre.
a is the distance from the vertex to the centre, when the vertices are on X-axis.
b is the distance from the vertex to the centre, when the vertices are on Y-axis.

Example

Solve:

• Its centre is at origin, so the unknowns are x² & y².
• According to its vertices, the hyperbola opens Up & Down.
• Since it’s up&down, the formula is y²/b² - x²/a² = 1.
• And the b² = √(6)² = 6
• According to the foci equation: then it should be 74 = a² + 6, and a² = 68.
• So the equation would be y²/6 - x²/68 = 1