$\textbf{Question : }$Given a segment AB in the first quadrant such that its length is L and is of the form :
$$-\frac{x}{L\cos\theta}+\frac{y}{L\sin \theta}=1$$ where $\theta\in\left(\dfrac{\pi}{2}, \pi\right)$. Take the $x$ and $y$ intercepts as point $A$ and $B$ then find the locus of a point $C$ in terms of $L$ such that $\Delta ABC$ is a equilateral triangle.
$\textbf{My Attempt : }$
First I reassigned $\theta$ to $\pi-\theta$ to avoid negative sign mistakes then taking $C$ as $(h,k)$ and applying the the equal distance condition we get $$(h-L\cos \theta)^2 + k^2=h^2+(k-L\sin\theta)^2$$ $$\frac{L}{2}\cos 2\theta = h\cos\theta-k\sin\theta$$ Now applying separately for one point $$(h-L\cos\theta)^2+k^2=L^2 \implies \cos \theta =\frac{h\pm\sqrt{L^2-k^2}}{L}$$ using this expression of $\cos\theta$ in the distance condition we get $$\frac{1}{2L}\left(2h^2-2k^2+L^2\pm4h\sqrt{L^2-k^2}\right) = h\left(\frac{h\pm\sqrt{L^2-k^2}}{L}\right)-k\sqrt{1-\left(\frac{h\pm\sqrt{L^2-k^2}}{L}\right)^2}$$ so technically I have found the locus but as you can see its quite a messy and long solution, so is there any better method for it...
Fig. 1: Case $L=1$. The locus is the blue elliptical arc, part of the ellipse obtained when the line segment $AB$ is authorized to move in the four quadrants.
Here is a solution using vectors.
I will take your renaming convention : $\theta = \angle OAB$ ($0 \le \theta \le \frac{\pi}{2}$) with
$$A(L \cos \theta,0) \ \ \ \text{and} \ \ \ B(0, L \sin \theta).$$
As the angle between $\vec{AC}$ and the $x$ axis is $2 \pi/3 -\theta$, one has :
$$\pmatrix{x\\y}=\vec{OC}=\vec{OA}+\vec{AC}=\pmatrix{L \cos \theta &+& L \cos(2 \pi/3 - \theta)\\&&L \sin(2 \pi/3 - \theta)}$$
which are the parametric equations of the locus.
Of course, they can be simplified:
$$\pmatrix{x\\y}=L\pmatrix{\frac12 \cos \theta &+& \frac{\sqrt{3}}{2}\sin\theta\\ \frac{\sqrt{3}}{2} \cos \theta &+&\frac12 \sin\theta}$$
$$\pmatrix{x\\y}=L\pmatrix{\frac12&\frac{\sqrt{3}}{2}\\ \frac{\sqrt{3}}{2}&\frac12}\pmatrix{\sin \theta\\ \cos \theta}$$
which is the image of an arc of circle by an affine transformation, i.e., an arc of ellipse.