Question

The hyperbola `x^2/a^2 - y^2/b^2` = 1 passes through the point `(3sqrt(5), 1)` and the length of its latus rectum is `4/3` units. The length of the conjugate axis is ______.

पर्याय

• 2 units

• 3 units

• 4 units

• 5 units

Answer 1

The hyperbola `x^2/a^2 - y^2/b^2` = 1 passes through the point `(3sqrt(5), 1)` and the length of its latus rectum is `4/3` units. The length of the conjugate axis is 4 units.

Explanation:

`x^2/a^2 - y^2/b^2` = 1

Hyperbola passes through `(3sqrt(5), 1)`

∴ `(3sqrt(5))^2/a^2 - 1/b^2` = 1

`45/a^2 - 1/b^2` = 1  ...(i)

Now length of latus rectum = `(2b^2)/a`

`\implies 4/3 = (2b^2)/a`

`\implies 2/3 = b^2/a`

`\implies` a = `(3b^2)/2`  ...(ii)

Putting the value of ‘a’ from equation (ii) in equation (i),

`\implies (45 xx 4)/(9b^4) - 1/b^2` = 1

`\implies 20/b^4 - 1/b^2` = 1

20 – b² = b⁴

b⁴ + b² – 20 = 0

b⁴ + 5b² – 4b² – 20 = 0

b² (b² + 5) – 4(b² + 5) = 0

(b² – 4) (b² + 5) = 0

b² = 4, b² = – 5

∴ b² = 4 `\implies` b = 2

Now length of conjugate axis = 2b = 2(2) = 4