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प्रश्न
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half of the distance between the foci is ______.
विकल्प
`4/3`
`4/sqrt(3)`
`2/sqrt(3)`
None of these
उत्तर
The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half of the distance between the foci is `2/sqrt(3)`.
Explanation:
Length of the latus rectum of the hyperbola
= `(2b^2)/a` = 8
⇒ b2 = 4a .......(i)
Distance between the foci = 2ae
Transverse axis = 2a
And Conjugate axis = 2b
∴ `1/2(2ae) = 2b`
⇒ ae = 2b
⇒ b = `(ae)/2` ......(ii)
⇒ `b^2 = (a^2e^2)/4`
⇒ `4a = (a^2e^2)/4` ......[From equation (i)]
⇒ 16 = ae2
∴ `a = 16/e^2`
Now b2 = a2(e2 – 1)
⇒ 4a = a2(e2 – 1)
⇒ `4/a = e^2 - 1`
⇒ `4/(16/e^2) = e^2 - 1`
⇒ `e^2/4 = e^2 - 1`
⇒ `e^2 - e^2/4` = 1
⇒ `(3e^2)/4` = 1
⇒ `e^2 = 4/3`
∴ e = `2/sqrt(3)`
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