Question

Select the coordinates that are perpendicular to the line

y=(-1/6)x-6 and goes through (2,-4)

There are two

a
(5,2)

b
(1,-10)

c
(4,5)

d
(3,2)

Answer 1

To find the points that are perpendicular to the given line \( y = -\frac{1}{6}x - 6 \) and go through the point \( (2, -4) \), we first need to determine the slope of the given line and the slope of the perpendicular lines.

The slope of the given line, \( m_1 \), is \( -\frac{1}{6} \). The slope of any line that is perpendicular to it, \( m_2 \), can be found using the negative reciprocal:

\[
m_2 = -\frac{1}{m_1} = -\frac{1}{-\frac{1}{6}} = 6
\]

Now, we can use the point-slope form of the equation of a line to find the equation of the line that is perpendicular to the original line and passes through the point \( (2, -4) \):

\[
y - y_1 = m(x - x_1)
\]

Substituting in the point \( (2, -4) \) and the slope \( 6 \):

\[
y - (-4) = 6(x - 2)
\]
\[
y + 4 = 6x - 12
\]
\[
y = 6x - 16
\]

Next, we will check which of the provided options lies on the line \( y = 6x - 16 \):

1. **For (5, 2)**:
\[
y = 6(5) - 16 = 30 - 16 = 14 \quad \text{(not equal to 2)}
\]

2. **For (1, -10)**:
\[
y = 6(1) - 16 = 6 - 16 = -10 \quad \text{(equal to -10)}
\]

3. **For (4, 5)**:
\[
y = 6(4) - 16 = 24 - 16 = 8 \quad \text{(not equal to 5)}
\]

4. **For (3, 2)**:
\[
y = 6(3) - 16 = 18 - 16 = 2 \quad \text{(equal to 2)}
\]

Thus, the coordinates that are perpendicular to the line \( y = -\frac{1}{6}x - 6 \) and go through the point \( (2, -4) \) are:

- (1, -10)
- (3, 2)

So the correct answers are **b** and **d**.