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If both the roots of the quadratic equat...

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

A

`(-oo,4)`

B

`[4,5]`

C

`(5,6)`

D

`(6,oo)`

Text Solution

Verified by Experts

The correct Answer is:
A
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)`
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