JEE Main 2026 · 6 Apr · Shift 1, Mathematics question 24

For the functions $f(\theta)=\alpha\tan^2\theta+\beta\cot^2\theta$, and $g(\theta)=\alpha\sin^2\theta+\beta\cos^2\theta$, $\alpha>\beta>0$, let $\displaystyle\min_{0<\theta<\frac\pi2}f(\theta)=\max_{0<\theta<\pi}g(\theta)$. If the first term of a G.P. is $\left(\dfrac\alpha{2\beta}\right)$, its common ratio is $\left(\dfrac{2\beta}\alpha\right)$ and the sum of its first 10 terms is $\dfrac mn$, $\gcd(m,n)=1$, then $m+n$ is equal to ________.