The value of $k$ for which the equation $x^{2}+2(k-1)x+k+5=0$ passes atleast one positive root are

The equation x^2 + 2(k-1)x + k+5 = 0 will have at least one positive root if either the smaller root is positive or the roots have opposite signs. Assuming the intended condition is exactly one positive root, the product of the roots (k+5) must be negative, giving k < -5. If the condition requires the smaller root to be positive with opposite signs ruled out by context, solving the standard boundary inequalities yields only k = -1 as the permissible value.

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