If both roots of the quadratic equation $x^2 - 2ax + a^2 - 1$ lie in $(-2, 2)$ then which of the following can be $[a]$ ?
$[a]$ denotes greatest Integer function of $a$
$$A. -1$$
$$B. 1$$
$$C. 2$$
$$D. 3$$
I have solve it using graphs:
The graph will intersect the x-axis somewhere between $-2$ and $2$. Hence we can conclude that $f(2)$ and $f(-2)$ will be greater than zero.
Now there will be two quadratic equation in $a$ and we will get $4$ values of $a$ I.e. $-3, -1 , 3 ,1$. From these values the answer should be opton $D.$ but the answer is given as option $A.$
Kindly help.
Observe that $$x^2 - 2ax + a^2 - 1=0$$ $$\implies x^2 - 2ax + a^2 = 1$$ $$\implies (x-a)^2=1$$ $$\implies x-a=\pm1$$ $$\implies x=a+1,a-1$$
Now it is given that both the roots lie between $-2$ and $2$.
Therefore, we have that $$-2 < a+1< 2$$ $$\implies -3 < a < 1\tag1$$ and $$-2 < a-1< 2$$ $$\implies -1 < a< 3 \tag2$$ This means from $(1)$ and $(2)$ that $-1 < a < 1$.
So $[a]$ should be either $0$ or $-1$.
Since $0$ is not in the option but $-1$ is, so the answer should be $(A)$.