Hyperbola: Foci

Hyperbola: Foci

How to find the foci of a hyperbola: formulas, examples, and their solutions.

Formula: Foci of x2/a2 - y2/b2 = 1

For the hyperbola x^2/a^2 - y^2/b^2 = 1, the foci are (+-c, 0) and c^2 = a^2 + b^2.

For the parabola x2/a2 - y2/b2 = 1,

the foci are (±c, 0)
and c2 = a2 + b2.

[c2 = a2 + b2] came from
the proof of the hyperbola formula.

Hyperbola - Proof of the formula x2/a2 - y2/b2 = 1

Example 1: Foci of x2/9 - y2/16 = 1

Find the foci of the given parabola. x^2/9 - y^2/16 = 1

Change the equation in standard form.

9 = 32
16 = 42

So x2/32 - y2/42 = 1.

a = 3
b = 4

So c2 = 32 + 42.

32 = 9
42 = 16

9 + 16 = 25

So c2 = 25.

Square root both sides.

Then c = ±5.

c = ±5
And the x2 term is (+).

So the foci are (±5, 0).

This is the graph of x2/32 - y2/42 = 1.

c = ±5
And the x2 term is (+).

So the foci are (±5, 0).

Formula: Foci of y2/b2 - x2/a2 = 1

For the hyperbola y^2/b^2 - x^2/a^2 = 1, the foci are (0, +-c) and c^2 = a^2 + b^2.

For the parabola y2/b2 - x2/a2 = 1,

the foci are (0, ±c)
and c2 = a2 + b2.

[c2 = a2 + b2] came from
the proof of the hyperbola formula.

Hyperbola - Proof of the formula y2/b2 - x2/a2 = 1

Example 2: Foci of x2 - y2 = -4

Find the foci of the given parabola. x^2 - y^2 = -4

Change the equation in standard form.

Divide both sides by -4.

Switch the order of the terms in the left side.

4 = 22

So y2/22 - x2/22 = 1.

a = 2
b = 2

So c2 = 22 + 22.

22 = 4

4 + 4 = 8

So c2 = 8.

Square root both sides.

Then c = ±√8.

8
= √22⋅2
= 2√2

Simplify a radical

So c = ±2√2.

c = ±2√2

And the y2 term of [y2/22 - x2/22 = 1] is (+).

So the foci are (0, ±2√2).

This is the graph of y2/22 - x2/22 = 1.

c = ±2√2
And the y2 term is (+).

So the foci are (0, ±2√2).