If the range of the function f(x) = 5-x/x^2-3x+2, x ≠ 1, 2, is (-∞, α] ∪ [β, ∞) , then α^2 + β^2 is equal to

Question

If the range of the function \(f(x)=\frac{5-x}{x^{2}-3 x+2}, x \neq 1,2,\) is \((-\infty, \alpha] \cup[\beta, \infty),\) then \(\alpha^{2}+\beta^{2}\) is equal to :

(1) 190

(2) 192

(3) 188

(4) 194  

Answer 1

Correct option is: (4) 194

\( y=\frac{5-x}{x^2-3 x+2} \)

\( x^2 y-3 x y+2 y=5-x \)

\( x^2 y+x(1-3 y)+2 y-5=0\)

For x to be real

D > 0 

\((1-3 y)^2-4 y(2 y-5)>0 \)

\( 1+9 y^2-6 y-8 y^2+20 y>0 \)

\( y^2+14 y+1>0 \)

\( y \in(-\infty, \alpha) \cup(\beta, \infty) \)

\( \alpha+\beta=-14 \)

\(\alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta \)

= 194