$f(x)=\left\{\begin{array}{cc}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}, & 0 < x < \frac{\pi}{2} \\ a-8, & x=\frac{\pi}{2} \\ (1+\mid \cot x)^{\frac{b}{a}|\tan x|}, & \frac{\pi}{2} < x < \pi\end{array}\right.$
Where $a, b \in Z$. If $f$ is continuous at $x=\frac{\pi}{2}$, then $\mathrm{a}^2+\mathrm{b}^2$ is equal to ..........
$\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{8}{7}\right)^{\frac{\tan 8 x}{\tan 7 x}}=\left(\frac{8}{7}\right)^0=1$
$RHL$ at $\mathrm{x}=\frac{\pi}{2}$
$\lim _{x \rightarrow \frac{\pi}{2}}(1+|\cot x|)^{\frac{b}{a}|\tan x|}$
$=\mathrm{e}^{\left.\lim _{\left.\mathrm{x} \rightarrow \frac{\pi}{2} \right\rvert\, \cot x} \mathrm{~b}\left|\frac{\mathrm{b}}{\mathrm{a}}\right| \tan x \right\rvert\,}=\mathrm{e}^{\frac{\mathrm{b}}{\mathrm{a}}}$
$\Rightarrow 1=\mathrm{a}-8=\mathrm{e}^{\frac{\mathrm{b}}{\mathrm{a}}}$
$\Rightarrow \mathrm{a}=9, \mathrm{~b}=0$
$\Rightarrow a^2+b^2=81$
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[Note: Here $z$ takes the values in the complex plane and $\operatorname{Im} z$ and $\operatorname{Re} z$ denote, respectively, the imaginary part and the real part of $z]$
| column-$I$ | column-$II$ |
| $(A)$ The set of points $z$ satisfying $|z-i| z||=|z+i| z||$ is contained in or equal to | $(p)$ an ellipse with eccentricity $\frac{4}{5}$ |
| $(B)$ The set of points $z$ satisfying $|z+4|+|z-4|=10$ is contained in or equal to | $(q)$ the set of points $z$ satisfying $\operatorname{Im} z=0$ |
| $(C)$ If $|\omega|=2$, then the set of points $z=\omega-1 / \omega$ is contained in or equal to | $(r)$ the set of points $z$ satisfying $|\operatorname{Im} z| \leq 1$ |
| $(D)$ If $|\omega|=1$, then the set of points $z=\omega+1 / \omega$ is contained in or equal to | $(s)$ the set of points $z$ satisfying $|\operatorname{Re} z| \leq 1$ |
| $(t)$ the set of points $z$ satisfying $|z| \leq 3$ |