If `alpha,beta(beta>alpha),` are the roots of `g(x)-a x^2+b x+c=0` and `f(x)` is an even function, then `int_alpha^betae/(e^(f((g(x))/(x-alpha)))+e^(f((g(x))/(x-beta))))=` `|b/(2a)|` (b) `(sqrt(b^2-4a c))/(|2a|)` `|b/a|` (d) none of these

Options

A. `|(b)/(a)|`
B. `|(b)/(2a)|`
C. `(sqrt(b^(2)-4ac))/(|2a|)`
D. none of these

Answer

C

Explanation

It is given that f(x) is an even function. Therefore, f(-x)=f(x) for all `x in R`
`rArr ax^(2)-bx+c=ax^(2)+bx+c` for all `x in R`
`rArr b=0 rArr(-b)/(a)=0 rArr alpha+beta=0 rArr beta=-alpha`
Also, `(f(-x))/(-x-alpha)=-(f(x))/(x-beta)"and"(f(-x))/(-x-beta)=(f(x))/(-x+alpha)=-(f(x))/(x-alpha)`
`:.f((f(-x))/(-x-a))=f(-(f(x))/(x-beta))=f((f(x))/(x-beta))`
and `f((f(-x))/(-x-beta))=f(-(f(x))/(x-alpha))=f((f(x))/(x-alpha))`
Let`I=underset(alpha)overset(beta)int(e^f((f(x))/(x-alpha)))/(e^f(f(x)/(x-alpha))+e^f((f(x))/(x-beta)))dx " "`....(ii)
Adding (i) and (ii), we obtain
`2I=underset(alpha)overset(beta)int dx=beta-alpha`
`rArr I=(1)/(2)sqrt((beta+alpha)^(2)-4alphabeta)=(sqrt(b^(2)-4ac))/(2|a|)`