A hyperbola is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant.

When the major axis is horizontal, the foci are at (-c,0) and at (0,c).

Let d_{1} be the distance from the focus at (-c,0) to the point at (x,y). Since
this is the distance between two points, we'll need to use the distance formula.

Similarly, d_{2} will involve the distance formula and will be the distance from
the focus at the (c,0) to the point at (x,y).

We can use the fact that the vertices are on the hyperbola to find out what the difference of the distances is.

If we take the vertex on the right, then d_{1} = c + a and d_{2} =
c - a.

d_{1} - d_{2} = ( c + a ) - ( c - a ) = c + a - c + a = 2a

Therefore, the constant is 2a and d_{1} - d_{2} = 2a for every
point on a hyperbola

Now that we know what the difference of the distances is, we can set about finding the equation of the hyperbola.

We start with d_{1} - d_{2} = 2a and substitute the formulas
for d_{1} and d_{2}. Because it is difficult to work with absolute values, we'll assume that
d_{1} is
greater than d_{2}, although the process doesn't change much if it's
the other way around.

We want to get rid of the radicals, so we'll move one term to the other side and then square both sides of the equation.

Let's multiply out the squared terms

If we move everything except for the square root term to the left side, a lot of this will cancel out and we'll get

Let's divide everything by 4 to get

Square both sides to get rid of the radical

Expand the (x-c)^{2} on the right hand side

Distribute the a^{2 }on the right side

Move all the variable terms to one side and the constants to the other.

Factor and x^{2} out of the first two terms on the left and an a^{2} out of the
right side.

Notice how there is an c^{2} - a^{2} on both sides. Let's
define b^{2} = c^{2} - a^{2 }and make the substitution
into the equation.

Finally, divide everything by a^{2}b^{2} so the right side is 1.

That gives us the standard form for a hyperbola with a transverse axis and
it also gives us the Pythagorean relationship c^{2} =
a^{2}+
b^{2}.

Return to notes on conic sections.