If both the roots of the quadratic equation `x^(2)-2kx+k^(2)+k-5=0` are less than 5, then k lies in the interval.

Options

A. [4, 5]
B. `(-oo, 4)`
C. `(6, oo)`
D. (5, 6]

Answer

B

Explanation

Let `f(x) = x^(2) - 2kx + k^(2) + k - 5`. If both roots of the given quadratic equation are less than 5, then we must have
(i) `D ge 0`
(ii) `f(5) gt 0`
(iii) Vertex of y = f(x) lies on the left side of (5, 0).
Now,
(i) `D ge 0 rArr 4k^(2) - 4(k^(2)+k-5) ge 0 rArr k - 5 le 0 rArr k le 5" "...(i)`
(ii) `f(5) gt 0 rArr k^(2) - 9k + 20 gt 0 rArr k lt 4 or k gt 5" "...(ii)`
(iii) Vertex of y = f(x) lies on the left hand side of (5, 0)
`rArr" "k lt 5" "...(iii)`
From (i), (ii) and (iii), we get `k in (-oo, 4)`.