Question

Select the coordinates that are on the line perpendicular to

y=x+2 and goes through (-2,-4)

There are two

a
(-1,-5)

b
(-3,6)

c
(0,0)

d
(1,-7)

Answer 1

To find the coordinates that are on the line perpendicular to \( y = x + 2 \) and that go through the point (-2, -4), we first need to determine the slope of the given line.

The slope of the line \( y = x + 2 \) is 1. The slope of a line perpendicular to this one will be the negative reciprocal of 1, which is -1.

Next, we can use the point-slope form of the line equation to find the equation of the line that is perpendicular to \( y = x + 2 \) and passes through the point (-2, -4).

The point-slope form is given by:

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

Where \( m \) is the slope, and \( (x_1, y_1) \) is the point (-2, -4). Substituting these values in:

\[
y - (-4) = -1(x - (-2))
\]

This simplifies to:

\[
y + 4 = -1(x + 2)
\]

Distributing on the right side:

\[
y + 4 = -x - 2
\]

Now, isolate \( y \):

\[
y = -x - 2 - 4
\]
\[
y = -x - 6
\]

Now we have the equation of the line that is perpendicular to \( y = x + 2 \) and passes through (-2, -4):

\[
y = -x - 6
\]

Next, we will check each of the given points to see which ones satisfy this equation.

1. For point (-1, -5):

\[
y = -(-1) - 6 = 1 - 6 = -5 \quad \text{(True)}
\]

2. For point (-3, 6):

\[
y = -(-3) - 6 = 3 - 6 = -3 \quad \text{(False)}
\]

3. For point (0, 0):

\[
y = -(0) - 6 = -6 \quad \text{(False)}
\]

4. For point (1, -7):

\[
y = -(1) - 6 = -1 - 6 = -7 \quad \text{(True)}
\]

The points that satisfy the equation of the perpendicular line \( y = -x - 6 \) are:

- a: (-1, -5)
- d: (1, -7)

Thus, the coordinates that are on the line perpendicular to \( y = x + 2 \) and goes through (-2, -4) are:

- a: (-1, -5)
- d: (1, -7)