How do you find the standard form of the equation of the hyperbola?

Answer 1

The standard form of a hyperbola in an x-y plot is (x-m)2/a2 - (y-n)2/b2 = +/- 1 (meaning either + or -), where (m,n) is the point of symmetry.

Let us take the case of +1 on the right-hand side.

Expanding the equation, we have

b2x2 - x * (2mb2) - a2y2 + y * (2na2) = a2b2 - b2m2 + a2n2

You have to match the coefficients of your equation to the ones above.

Let us assume your equation is x2 - y2 = 1, which will be trivial to do.

Matching the coefficients, you have m = 0 = n and a = +/-1 = b (insufficient information to determine + or -).

Another simple example for your equation: 9x2 - 18x - 4y2 - 8y = 31.

Matching coefficients exactly, we have

b2 = 9 -- (1)

-2mb2 = -18 -- (2)

-a2 = -4 -- (3)

2na2 = -8; and -- (4)

a2b2 - b2m2 + a2n2 = 31 -- (5)

(1) is simplified to b = +/-3 (no way knowing + or -)

(3) is simplified to a = +/-2

(1) substituted into (2), (2) becomes m = 1

(3) substituted into (4), (4) becomes n = -1

Substituting (1-4) into 5, we have the LHS = 36 - 9 + 4 = 31 = RHS.

For our convenience, if +/- cannot be determined, take the + for the equation coefficients. Hence, a = 2 and b = 3.

Q.E.D.