Explanation

(a)
$\sqrt{x^{2} + (y - 1)^{2}} = k\sqrt{x^{2} + (y + 1)^{2}}$
$x^{2} + y^{2} - 2y + 1 = k^{2} + \left( x^{2} + y^{2} + 2y + 1 \right)$
$+ 2k\sqrt{x^{2} + (y + 1)^{2}}$
$\Rightarrow \ - 4y - k^{2} = 2k\sqrt{x^{2} + (y + 1)^{2}}$
$\Rightarrow \ 16y^{2} + k^{4} + 8yk^{2} = 4k^{2}\left( x^{2} + y^{2} + 2k + 1 \right)$ [squaring]
$\Rightarrow \ 4x^{2}k^{2} + \left( 4k^{2} - 16\ \right)\ y^{2} = k^{4} - 4k^{2}$
To represent an equation of hyperbola, the coefficient of either $x^{2}$ or $y^{2}$ is negative
But coefficient of $x^{2}$ cannot be negative, so we take the coefficient of $y^{2}$
$4k^{2} - 16 < 0$
$\Rightarrow \ k^{2} \leq 4$
$\Rightarrow \ - \ 2 < k < 2$
As the given equation $k$ cannot be negative
$0 < k < 2$