If the domain of the function log5(18x-x^2-77) is (α, β) and the domain of the function log (x-1)

Question

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

(1) 195

(2) 174

(3) 186

(4) 179

Answer 1

Correct option is (3) 186   

\( f_1(x)=\log _5\left(18 x-x^2-77\right)\) 

\( \therefore \ 18 x-x^2-77>0 \)    

\( x^2-18 x+77<0 \)

\( x \in(7,11) \)

\( \alpha=7, \beta=11 \)

\( f_2(x)=\log _{(x-1)}\left(\frac{2 x^2+3 x-2}{x^2-3 x-4}\right)\)

\(x>1, x-1 \neq 1, \frac{2 x^2+3 x-2}{x^2-3 x-4}>0\)

\(x>1, x \neq 2, \frac{(2 x-1)(x+2)}{(x-4)(x+1)}>0\)

\(x>1, x \neq 2\), 

\(\therefore \ x \in(4, \infty) \)   

\(\therefore \gamma=4 \)   

\(\therefore \quad \alpha^2+\beta^2+\gamma^2=49+121+16 \)   

= 186