Question

The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half of the distance between the foci is ______.

Options

• `4/3`

• `4/sqrt(3)`

• `2/sqrt(3)`

• None of these

Answer 1

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)`