Coordinate Geometry: JEE Main 2022 Mathematics question

For $\mathrm{t} \in(0,2 \pi)$, if $\mathrm{ABC}$ is an equilateral triangle with vertices $\mathrm{A}(\sin t,-\cos \mathrm{t}), \mathrm{B}(\operatorname{cost}, \sin t)$ and $C(a, b)$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $\left(a^{2}-b^{2}\right)$ is equal to :