Question

For the equation $x^2$ - 2ax + $a^2$ - 1 = 0, the values of 'a' for which 3 lies in between the roots of given equation is

• A. (-, 3) (4, )
• B. (-, 2) (4, )
• C. (2, 4)
• D. [3, 4]

Answer 1

The correct option is C

(2, 4)

For roots to be real,

$b^2$ - 4ac > 0

$\Rightarrow$ 4$a^2$ - 4($a^2$ - 1) > 0

1 > 0 [Always true]

Here a > 0,

So, the given quadratic expression will have minimum value at x = -b/2A = a

For 3 to lie between the roots, f(3) < 0.

Hence f(3) = 9 - 6a + $a^2$ - 1 < 0

$a^2$ - 6a + 8 < 0

$a^2$ - 4a - 2a + 8 < 0

a(a - 4) - 2(a - 4) < 0

$\Rightarrow$ 2 < a < 4 ------------ (1)

Hence a $ϵ$ (2, 4)