Sure, let's solve this question step-by-step:
Given points are [tex]\(P \left(2a^2 - 2, 5 \right) \)[/tex] and [tex]\( Q (1, 3a - 4) \)[/tex].
The slope of the line passing through two points [tex]\((x_1, y_1) \)[/tex] and [tex]\((x_2, y_2) \)[/tex] is given by:
[tex]\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
In this case, [tex]\((x_1, y_1) = (2a^2 - 2, 5) \)[/tex] and [tex]\((x_2, y_2) = (1, 3a - 4) \)[/tex].
Since the slope is given to be zero, the numerator of the slope equation must be zero:
[tex]\[
\frac{(3a - 4) - 5}{1 - (2a^2 - 2)} = 0
\][/tex]
Simplifying this, we get the equation:
[tex]\[
(3a - 4) - 5 = 0
\][/tex]
This can be further simplified as:
[tex]\[
3a - 4 - 5 = 0
\][/tex]
[tex]\[
3a - 9 = 0
\][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[
3a = 9
\][/tex]
[tex]\[
a = 3
\][/tex]
Therefore, the possible value of [tex]\(a\)[/tex] is:
[tex]\[
\boxed{3}
\][/tex]
So, the correct answer is:
C. 3
