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

Answer

A

Explanation

Let `f(x)=x^(2)-2kx+k^(2)+k` then `alpha`
Consider the following cases:

Case I `Dge0`
`implies4k^(2)-4.1(ik^(2)+k-5)ge0`
`implies-4(k-5)ge0`
`impliesk-5le0`
`implieskle5` or `kepsilon(-oo,5]`
Case II x-Coordinate of vertex `xlt5`
`implies(2k)/2lt5`
`impliesklt5` or `k epsilon (-oo,5)`
Case III `f(5)gt0`
`implies2-10k+k^(2)+k-5gt0`
`impliesk^(2)-9k+20gt0`
`implies(k-4)(k-5)gt0` or `k epsilon (-oo,4)uu(5,oo)`
Combining all cases, we get
`k epsilon (-oo,4)`