Answer:
To find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\), we use the distance formula:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Given that the distance between \((3,-1)\) and \((k-5)\) is \( \sqrt{52} \) units, we can plug in the values:
\[ \sqrt{(k-5-3)^2 + (-1 - (-1))^2} = \sqrt{52} \]
\[ \sqrt{(k-8)^2 + 0} = \sqrt{52} \]
\[ (k-8)^2 = 52 \]
Now, let's solve for \(k\):
\[ k - 8 = \pm \sqrt{52} \]
\[ k = 8 \pm \sqrt{52} \]
\[ k = 8 \pm 2\sqrt{13} \]
So, the possible values of \(k\) are \( 8 + 2\sqrt{13} \) and \( 8 - 2\sqrt{13} \).
