The function`f(x)={{:(,(x^(2))/(a),0 le x lt 1),(,a,1le x lt sqrt2),(,(2b^(2)-4b)/(x^(2)),sqrt2 le x lt oo):}` is a continuous for `0 le x lt oo`. Then which of the following statements is correct?

Options

A. The number of all possible ordered pairs (a,b) is 3
B. The number of all possible ordered pairs (a,b) is 4
C. The product of all possible pairs ,b is -1
D. The product of all possible values of b is 1

Answer

A, C

Explanation

It is given that f(x) is continuous on `[0,oo)`. So, it is continuous at x=1 and `x=sqrt2`
`therefore underset(x to 1^(-))lim f(x)=underset(x to sqrt2^(+))lim f(x)=f(sqrt2)`
`Rightarrow underset(x to 1^(-))lim (x^(2))/(a)=underset(xto 1)lim a and underset(x to sqrt2^(-))lim a=underset(x to sqrt2^(+))lim (2b^(2)-4b)/(x^(2))`
`Rightarrow (1)/(a)=a and a=(2b^(2)-4b)/(2)`
`Rightarrow a^(2)=1 and b^(2)-2b=a`
`Rightarrow a=pm1 and b^(2)-2b=a`
`Rightarrow a=1,b^(2)-2ab-1=0 or, a=-1,b^(2)-2b+1=0`
`Rightarrow a=1,b=1 pmsqrt2 or, a=-1,b=1`
`Rightarrow (a,b)=(1,1+sqrt2),(1,1-sqrt2),(-1,1)`