which point on the y - axis lies on the line that passes through point c and is perpendicular to line ab…

Explanation:

Step1: Find the slope of line AB

Let (A(-2,4)) and (B(2, - 8)). The slope formula is (m=\frac{y_2 - y_1}{x_2 - x_1}). So, (m_{AB}=\frac{-8 - 4}{2-(-2)}=\frac{-12}{4}=- 3).

Step2: Find the slope of the perpendicular line

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the line perpendicular to (AB) be (m). Then (-3m=-1), so (m = \frac{1}{3}).

Step3: Use the point - slope form to find the equation of the line passing through point C

Let (C(6,4)). The point - slope form of a line is (y - y_1=m(x - x_1)). Substituting (m=\frac{1}{3}), (x_1 = 6) and (y_1 = 4) gives (y - 4=\frac{1}{3}(x - 6)).

Step4: Simplify the equation

Expand the right - hand side: (y-4=\frac{1}{3}x - 2). Then (y=\frac{1}{3}x+2).

Step5: Find the y - intercept

The line intersects the (y) - axis when (x = 0). Substitute (x = 0) into (y=\frac{1}{3}x + 2), we get (y=2). So the point on the (y) - axis is ((0,2)).

Answer:

C. ((0,2))