Question

Find the equation of the hyperbola referred to its principal axes:

which passes through the points (6, 9) and (3, 0)

Answer 1

Let the required equation of hyperbola be `x^2/"a"^2 - y^2/"b"^2` = 1.   ...(i)

The hyperbola passes through the points (6, 9) and (3, 0).

∴ Substituting x = 6 and y = 9 in (i), we get

`6^2/"a"^2 - 9^2/"b"^2` = 1

∴ `36/"a"^2 - 81/"b"^2` = 1     ...(ii)

Substituting x = 3 and y = 0 in (i), we get

`3^2/"a"^2 - 0^2/"b"^2` = 1

∴ `9/"a"^2 - 0` = 1

∴ a² = 9

Substituting a² = 9 in (ii), we get

`36/9 - 81/"b"^2` = 1

∴ `81/"b"^2 = 36/9 - 1`

∴ `81/"b"^2` = 4 – 1 = 3

∴ b² = `81/3` = 27

∴ The required equation of hyperbola is `x^2/9 - y^2/27` = 1.