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)

Answer 1

Answer: C (0,2)

Explanation:

From the given figure in the question it is observed that the line AB passes through the points and .

The coordinate for the point C is .

Step 1: Obtain the slope of the line AB.

The slope of a line which passes through the points and is calculated as follows:

(1)

It is given that the line AB passes through the points and .

To obtain the slope for the line AB substitute for , for , for and for in equation (1).

Therefore, the slope of the line AB is .

Consider the slope of AB as, so, .

Step2: Obtain the slope of the perpendicular line.

The slope of line AB is .

Consider a line which is perpendicular to the line AB passing through the point C. Assume the slope of the perpendicular line as .

The product of slope of two mutually perpendicular lines is always equal to .

The equation formed for the slope is as follows:

Substitute the value of in the above equation.

Therefore, the slope of the perpendicular line is .

Step3: Obtain the equation of the perpendicular line.

The slope for perpendicular line is and the line passes through the point C. The coordinate for the point C are .

The point slope form of a line is as follows:

Substitute for , for and for in the above equation.

Therefore, the equation of the perpendicular line is .

Option 1:

In option 1 it is given that the line perpendicular to AB passes through the point .

The equation of the line which is perpendicular to AB is .

Substitute for in the above equation.

From the above calculation it is concluded that the line passes through the point .

This implies that option 1 is incorrect.

Option 2:

In option 2 it is given that the line perpendicular to AB passes through the point .

The equation of the line which is perpendicular to AB is .

Substitute for in the above equation.

From the above calculation it is concluded that the line passes through the point .

This implies that option 2 is incorrect.

Option 3:

In option 3 it is given that the line perpendicular to AB passes through the point .

The equation of the line which is perpendicular to AB is .

Substitute for in the above equation.

From the above calculation it is concluded that the line passes through the point .

This implies that option 3 is correct.

Option 4:

In option 4 it is given that the line perpendicular to AB passes through the point .

The equation of the line which is perpendicular to AB is .

Substitute for in the above equation.

From the above calculation it is concluded that the line passes through the point .

This implies that option 4 is incorrect.

Therefore, the line perpendicular to the line AB passes through the point.

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Answer 2

If we need our line to pass through point C, then we have to use the x and coordinates of point C in our new equation. If that line is to be perpendicular to AB, we also need to find the slope of AB and then take its opposite reciprocal. First things first. Point C lies at (6, 4) so we will use x = 6 and y = 4 in our equation in a bit. The coordinates of A are (-2, 4) and the coordinates of B are (2, -8) so the slope between them is
`m= (-8-4)/(2-(-2))` which is -3. The opposite reciprocal of -3 is 1/3. That's the slope we will use along with the points from C to write the new equation. We will do this by plugging in x, y, and m (slope) into the slope-intercept form of a line and solve for b.
`4= (1)/(3) (6)+b` and 4 = 2 + b. So b = 2. That's the y-intercept, the point on the y axis where the line goes through when x is 0. Therefore, the point you're looking for is (0, 2).