The question is incomplete! Complete question along with answer and step by step explanation is provided below.
Question:
The given line passes through the points (-4, -3) and (4, 1).
What is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point (-4, 3)?
a) y - 3 = -2(x + 4)
b) y - 3= - (x + 4)
c) y - 3 = (x + 4)
d) y - 3 = 2(x + 4)
Answer:
The equation of the line that is perpendicular to the given line and passes through the point (-4, 3) is
a) y - 3 = -2(x +4)
Step-by-step explanation:
First of all, we will find the slope of the given line.
We are given that the line passes through the points (-4, -3) and (4, 1)
![(x_1, y_1) = (-4,-3) \\\\(x_2, y_2) = (4,1) \\\\](https://tex.z-dn.net/?f=%28x_1%2C%20y_1%29%20%3D%20%28-4%2C-3%29%20%5C%5C%5C%5C%28x_2%2C%20y_2%29%20%3D%20%284%2C1%29%20%5C%5C%5C%5C)
The slope of the equation is given by
![$ m_1 = \frac{y_2 - y_1 }{x_2 - x_1} $](https://tex.z-dn.net/?f=%24%20m_1%20%3D%20%5Cfrac%7By_2%20-%20y_1%20%7D%7Bx_2%20-%20x_1%7D%20%20%24)
![m_1 = \frac{1 -(-3) }{4 -(-4)} \\\\m_1 = \frac{1 + 3 }{4 + 4} \\\\m_1 = \frac{4 }{8} \\\\m_1 = \frac{1 }{2} \\\\](https://tex.z-dn.net/?f=m_1%20%3D%20%5Cfrac%7B1%20-%28-3%29%20%7D%7B4%20-%28-4%29%7D%20%20%5C%5C%5C%5Cm_1%20%3D%20%5Cfrac%7B1%20%2B%203%20%7D%7B4%20%2B%204%7D%20%20%5C%5C%5C%5Cm_1%20%3D%20%5Cfrac%7B4%20%7D%7B8%7D%20%20%5C%5C%5C%5Cm_1%20%3D%20%5Cfrac%7B1%20%7D%7B2%7D%20%20%5C%5C%5C%5C)
Recall that the slopes of two perpendicular lines are negative reciprocals of each other.
![$ m_2 = - \frac{1}{m_1} $](https://tex.z-dn.net/?f=%24%20m_2%20%3D%20-%20%5Cfrac%7B1%7D%7Bm_1%7D%20%24)
So the slope of the other line is
![m_2 = - 2](https://tex.z-dn.net/?f=m_2%20%3D%20-%202)
Now we can find the equation of the line that is perpendicular to the given line and passes through the point (-4, 3)
The point-slope form is given by,
![y - y_1 = m(x -x_1)](https://tex.z-dn.net/?f=y%20-%20y_1%20%3D%20m%28x%20-x_1%29)
Substitute the value of slope and the given point
![y - 3 = -2(x -(-4) \\\\y - 3 = -2(x +4)](https://tex.z-dn.net/?f=y%20-%203%20%3D%20-2%28x%20-%28-4%29%20%5C%5C%5C%5Cy%20-%203%20%3D%20-2%28x%20%2B4%29)
Therefore, the correct option is (a)
y - 3 = -2(x + 4)