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Question
Find the equation of the hyperbola referred to its principal axes:
which passes through the points (6, 9) and (3, 0)
Solution
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.
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