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Question
Answer the following:
Find the equation of the hyperbola in the standard form if Length of conjugate axis is 5 and distance between foci is 13.
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 = 5
∴ 2b = 5
∴ b = `5/2`
∴ b2 = `25/4`
Distance between foci = 2ae
Given, distance between foci = 13
∴ 2ae = 13
∴ ae = `13/2`
∴ a2e2 = `169/4`
∴ Now, b2 = a2(e2 – 1)
∴ b2 = a2e2 – a2
∴ `25/4 = 169/4` – a2
∴ a2 = `169/4 - 25/4`
∴ a2 = `144/4` = 36
∴ The required equation of hyperbola is
`x^2/36 - y^2/(25/4)` = 1,
i.e., `x^2/36 - (4y^2)/25` = 1
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