Question

The locus of the point of intersection of lines `sqrt(3)x - y - 4sqrt(3)k` = 0 and `sqrt(3)kx + ky - 4sqrt(3)` = 0 for different value of k is a hyperbola whose eccentricity is 2.

Options

• True

• False

Answer 1

This statement is True.

Explanation:

The given equations are

 `sqrt(3)x - y - 4sqrt(3)k` = 0   ......(i)

`sqrt(3)kx + ky - 4sqrt(3)` = 0  ......(ii)

From equation (i) we get

`4sqrt(3)k = sqrt(3)x - y`

∴ `k = (sqrt(3)x - y)/(4sqrt(3))`

Putting the value of k in equation (ii), we get

`sqrt(3)[(sqrt(3)x - y)/(4sqrt(3))]x + [(sqrt(3)x - y)/(4sqrt(3))]y - 4sqrt(3)` = 0

⇒ `((sqrt(3)x - y)/4)x + ((sqrt(3)x - y)/(4sqrt(3)))y - 4sqrt(3)` = 0

⇒ `((3x - sqrt(3)y)x + (sqrt(3)x - y)y - 48)/(4sqrt(3))` = 0

⇒ `3x^2 - sqrt(3)xy + sqrt(3)xy - y^2 - 48` = 0

⇒ `3x^2 - y^2` = 48

⇒ `x^2/16 - y^2/48` = 1 which is a hyperbola.

Here a² = 16, b² = 48

We know that b² = a²(e² – 1)

⇒ 48 = 16(e² – 1)

⇒ 3 = e² – 1

⇒ e² = 4

⇒ e = 2