If the domain of the function $\log _{5}\left(18 x-x^{2}-77\right)$ is $(\alpha, \beta)$ and the dom

Question

If the domain of the function $\log _{5}\left(18 x-x^{2}-77ight)$ is $(\alpha, \beta)$ and the domain of the function $\log _{(x-1)}\left(\frac{2 x^{2}+3 x-2}{x^{2}-3 x-4}ight)$ is $(\gamma, \delta)$, then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is equal to :\

Professor PC Thomas Classes · Answer

(c): Let $f_{1}(x)=\log _{5}\left(18 x-x^{2}-77 ight)$\ $ herefore \quad 18 x-x^{2}-77>0$\ $\Rightarrow \quad x^{2}-18 x+77<0$\ $\Rightarrow \quad(x-11)(x-7)<0$\ $\Rightarrow \quad x \in(7,11) herefore \alpha=7, \beta=11$\ Also, let $f_{2}(x)=\log _{(x-1)}\left(\frac{2 x^{2}+3 x-2}{x^{2}-3 x-4} ight)$\ $ herefore \quad x-1>0 \Rightarrow x>1, x-1 eq 1 \Rightarrow x eq 2$, $\frac{2 x^{2}+3 x-2}{x^{2}-3 x-4}>0$\ $\Rightarrow \frac{(2 x-1)(x+2)}{(x-4)(x+1)}>0$\ $\Rightarrow \quad x \in(4, \infty) herefore \gamma=4$\ Now, $\alpha^{2}+\beta^{2}+\gamma^{2}=49+121+16=186$\