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If the equation x^(log(a)x^(2))=(x^(k-2)...

If the equation `x^(log_(a)x^(2))=(x^(k-2))/a^(k),a ne 0`has exactly one solution for x, then the value of k is/are

A

`6+4sqrt2`

B

`2+6sqrt3)`

C

`6-4sqrt2`

D

`2-6sqrt3`

Text Solution

Verified by Experts

The correct Answer is:
A, C
` (log_(a)x^(2)) log_(a)x =(k-2)log_(a) x - k`
(taking log on base a )
Let ` log_(a) x = t`, we get
` 2t^(2) - (k-2) t+k = 0`
Putting D= 0 (has only one solution), we have
` (k-2)^(2) - 8k=0`
` or k^(2) - 12k+4 = 0`
` or k=(12 pm sqrt(128))/2`
` or k = 6 pm 4 sqrt2`
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