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Question
Answer the following:
Find the equation of the hyperbola in the standard form if length of the conjugate axis is 3 and distance between the foci is 5.
Solution
Let the required equation of hyperbola be
`x^2/"a"^2 - y^2/"b"^2` = 1
Length of conjugate axis = 2b
Given, length of conjugate axis = 3
∴ 2b = 3
∴ b = `3/2`
∴ b2 = `9/4`
Distance between foci = 2ae
Given, distance between foci = 5
∴ 2ae = 5
∴ ae = `5/2`
∴ a2e2 = `25/4`
Now, b2 = a2(e2 – 1)
∴ b2 = a2e2 – a2
∴ `9/4 = 25/4` – a2
∴ a2 = `25/4 - 9/4`
∴ a2 = `16/4` = 4
∴ The required equation of hyperbola is
`x^2/4 - y^2/((9/4))` = 1,
i.e., `x^2/4 - (4y^2)/9` = 1
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