For some $\theta \in\left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2-y^2 \sec ^2 \theta=8$ be $e_1$ and $l_1$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^2 \sec ^2 \theta+y^2=6$ be $e_2$ and $l_2$, respectively. If $e_1^2=e_2^2\left(\sec ^2 \theta+1\right)$, then $\left(\frac{l_1 l_2}{e_1 e_2}\right) \tan ^2 \theta$ is equal to

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $\mathbb{R}-(a, b)$, then $a^2+b^2$ is equal to

$\_\_\_\_$ .

If $k=\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{2}{3}\right)\right)+\tan \left(\frac{1}{2} \sin ^{-1}\left(\frac{2}{3}\right)\right)$, then

the number of solutions of the equation $\sin ^{-1}(k x-1)=\sin ^{-1} x-\cos ^{-1} x$ is $\_\_\_\_$.

$$ \text { The value of } \sum\limits_{r=1}^{20}\left(\left|\sqrt{\pi\left(\int\limits_0^r x|\sin \pi x| d x\right)}\right|\right) \text { is } $$