Choose the correct answer from the given alternatives and write the letter of your choice on the provided line.

If the slope of the line passing through the points [tex]P (2a^2 - 2, 5)[/tex] and [tex]Q (1, 3a - 4)[/tex] is zero, then the possible value(s) of [tex]a[/tex] is (are):

A. -3 and 3
B. [tex]$\pm 1$[/tex]
C. 3
D. 1



Answer :

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