Step-by-step explanation:
[tex] \sqrt{(k - 2)^{2} + {(9 + 3)}^{2} } = 13 \\ \\ \therefore \: \sqrt{k^{2} - 2 \times 2k + 2^{2} + {(12)}^{2} } = 13 \\ \\ \therefore \: \sqrt{k^{2} - 4k + 4 + 144 } = 13\\ \\ \therefore \: \sqrt{k^{2} - 4k + 148 } = 13 \: \\ \: \: \: \: \: \: squaring \: both \: sides \\ \\ k^{2} - 4k + 148 = 169 \\ \\ \therefore \: k^{2} - 4k + 148 - 169 = 0\\ \\ \therefore \: k^{2} - 4k - 21 = 0\\ \\ \therefore \: k^{2} - 7k + 3k - 21 = 0\\ \\ \therefore \: k(k - 7) + 3(k - 7) = 0 \\ \\ \therefore \: (k - 7) (k+ 3) = 0 \\ \\ \therefore \: (k - 7) = 0 \: or \: (k+ 3) = 0 \\ \\ \therefore \: k = 7 \: or \: k = - 3 \\ \\ \huge \purple{ \boxed{\therefore \:k = \{ - 3, \: \: 7 \}}}[/tex]
