Question

The hyperbola `x^2/a^2 - y^2/b^2` = 1 passes through the point of intersection of the lines `x - 3sqrt(5)y` = 0 and `sqrt(5)x - 2y` = 13 and the length of its latus rectum is `4/3` units. The coordinates of its focus are ______.

Options

• `(±2sqrt(10), 1)`

• `(±3sqrt(10), 0)`

• `(±2sqrt(10), 0)`

• `(±3sqrt(10), 1)`

Answer 1

The hyperbola `x^2/a^2 - y^2/b^2` = 1 passes through the point of intersection of the lines `x - 3sqrt(5)y` = 0 and `sqrt(5)x - 2y` = 13 and the length of its latus rectum is `4/3` units. The coordinates of its focus are `underlinebb((±2sqrt(10), 0)`.

Explanation:

Point of intersection, `P(3sqrt(5), 1)` [Solving given lines]

Hyperbola: `x^2/a^2 - y^2/b^2` = 1, passes through 'P'

∴ `45/a^2 - 1/b^2` = 1  ...(i)

Also, `(2b^2)/a = 4/3` [Length of L.R.]

b² = `(2a)/3`  ...(ii)

And b² = a²(e² – 1)

b² + a² = a²e²  ...(iii)

From (i) and (ii)

2a² + 3a – 90 = 0

2a(a – 6) + 15(a – 6) = 0

a = 6 or a = `(-15)/2`

∴ b² = `(2 xx 6)/3` = 4

Putting the value of a and b in (iii)

a²e² = a² + b²

= 36 + 4

= 40

ae = `±2sqrt(10)`

Hence focus `(±2sqrt(10), 0)`