The question is incomplete! Complete question along with answer and step by step explanation is provided below.
Question:
Which point on the y-axis lies on the line that passes through point C and is perpendicular to line AB?
A. (-6, 0)
B. (0, -6)
C. (0, 2)
D. (2, 0)
The graph of the question is attached.
Answer:
The point is (x, y) = (0, 2)
The correct option is C.
Therefore, the point (0, 2) on the y-axis lies on the line that passes through point C and is perpendicular to line AB.
Step-by-step explanation:
From the given graph, the points A and B are
[tex](x_1, y_1) = (-2, 4) \\\\(x_2, y_2) = (2,-8) \\\\[/tex]
The slope of the equation is given by
[tex]m_1 = \frac{-8 - 4 }{2 -(-2)} \\\\ m_1 = \frac{-12 }{2+2} \\\\m_1 = \frac{-12 }{4} \\\\m_1 = -3 \\\\[/tex]
We know that the slopes of two perpendicular lines are negative reciprocals of each other.
[tex]m_2 = - \frac{1}{m_1}[/tex]
So the slope of the other line is
[tex]m_2 = \frac{1 }{3} \\\\[/tex]
Now we can find the equation of the line that is perpendicular to the line AB and passes through the point C.
From the graph, the coordinates of point C are
[tex](x_1, y_1) = (6, 4)[/tex]
The point-slope form is given by,
[tex]y - y_1 = m(x -x_1)[/tex]
Substitute the value of slope and the coordinates of point C
[tex]y - 4 = \frac{1 }{3} (x - 6)\\\\[/tex]
To get the y-intercept, substitute x = 0
[tex]y - 4 = \frac{1 }{3} (0 - 6) \\\\y - 4 = \frac{-6 }{3}\\\\y - 4 = -2\\\\y = 4 -2 \\\\y = 2 \\\\[/tex]
So, the point is
[tex](x, y) = (0, 2)[/tex]
The correct option is C.
Therefore, the point (0, 2) on the y-axis lies on the line that passes through point C and is perpendicular to line AB.
