Question

which point on the y - axis lies on the line that passes through point c and is perpendicular to line ab? (-6,0) (0,-6) (0,2) (2,0)

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. \((0,2)\)