Question

Sol. [Key Concept: Equilateral side sliding on the axes: parametrise the rotating segment to get the apex locus.]

$h = 2cos\theta + 2cos\left( 120^{\circ} - \theta \right)$ $= cos\theta + \sqrt{3}sin\theta$ $(1)$

$$\begin{array}{r} \ \# \\ k\ = 0 + 2sin\left( 120^{\circ} - \theta \right) = \sqrt{3}cos\theta + sin\theta\#(2)\# \\ \ \end{array}$$

squaring and adding eqs. (1) and (2), we get

$$h^{2} + k^{2} = 4 + 4\sqrt{3}sin\theta cos\theta$$

Multiplying eqs. (1) and (2), we get

$$hk = \sqrt{3} + 4sin\theta cos\theta$$

$$\Rightarrow \ \left( h^{2} + k^{2} \right) - 4 = \sqrt{3}hk - 3 $$$x^{2} + y^{2} = \sqrt{3}xy + 1$ is the required locus.