Option D:
The equation of a line in point-slope form is y = 3x – 7.
Solution:
Take any two points on the line given in the graph.
Let the points be (–4, 4) and (2, 2).
[tex]x_1=-4, y_1=4, x_2=2, y_2=2[/tex]
Slope of the given line:
[tex]m_1=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]m_1=\frac{2-4}{2-(-4)}[/tex]
[tex]m_1=\frac{-2}{6}[/tex]
[tex]m_1=\frac{-1}{3}[/tex]
If two lines are perpendicular, then the product of their slopes are –1.
[tex]\Rightarrow m_1\times m_2=-1[/tex]
[tex]\Rightarrow \frac{-1}{3} \times m_2=-1[/tex]
Multiply by 3 on both sides of the equation.
[tex]\Rightarrow- m_2=-3[/tex]
Multiply by –1 on both sides of the equation.
⇒ [tex]m_2=3[/tex]
Perpendicular line passes through the point (2, –1).
Here, [tex]x_1=2, y_1=-1[/tex].
Using point-slope form:
[tex]y-y_1=m(x-x_1)[/tex]
[tex]y-(-1)=3(x-2)[/tex]
[tex]y+1=3x-6[/tex]
Subtract 1 on both sides of the equation.
[tex]y=3x-7[/tex]
The equation of a line in point-slope form is y = 3x – 7.
Therefore option D is the correct answer.
