To determine the values of [tex]\( k \)[/tex] for which the distance between the points [tex]\((-3, k)\)[/tex] and [tex]\((2, 0)\)[/tex] is [tex]\(\sqrt{34}\)[/tex], we will use the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here,
[tex]\[ x_1 = -3, \][/tex]
[tex]\[ y_1 = k \][/tex]
[tex]\[ x_2 = 2, \][/tex]
[tex]\[ y_2 = 0 \][/tex]
[tex]\[ d = \sqrt{34} \][/tex]
We substitute the values into the distance formula:
[tex]\[ \sqrt{34} = \sqrt{(2 - (-3))^2 + (0 - k)^2} \][/tex]
Simplify inside the square root:
[tex]\[ \sqrt{34} = \sqrt{(2 + 3)^2 + (0 - k)^2} \][/tex]
[tex]\[ \sqrt{34} = \sqrt{5^2 + (-k)^2} \][/tex]
[tex]\[ \sqrt{34} = \sqrt{25 + k^2} \][/tex]
Next, remove the square roots by squaring both sides of the equation:
[tex]\[ 34 = 25 + k^2 \][/tex]
To isolate [tex]\( k^2 \)[/tex], subtract 25 from both sides:
[tex]\[ 34 - 25 = k^2 \][/tex]
[tex]\[ 9 = k^2 \][/tex]
Now, solve for [tex]\( k \)[/tex] by taking the square root of both sides:
[tex]\[ k = \pm \sqrt{9} \][/tex]
[tex]\[ k = \pm 3 \][/tex]
Therefore, the values of [tex]\( k \)[/tex] that satisfy the condition are [tex]\( k = 3 \)[/tex] and [tex]\( k = -3 \)[/tex].
Thus, the correct answer is:
A. 3 or -3
