Option B: [tex]y=-\frac{1}{2} x-4[/tex] is the equation of the line.
Explanation:
Given that the equation of the line is perpendicular to [tex]y = 2x + 8[/tex] and passes through the point [tex](8,-8)[/tex]
We need to determine the equation of the line.
The equation of the line can be determined using the formula,
[tex]y-y_1=m(x-x_1)[/tex]
First, we shall find the value of m.
From the equation [tex]y = 2x + 8[/tex], the slope is 2.
Since, the lines are perpendicular, the slope is given by
[tex]m=-\frac{1}{2}[/tex]
Thus, the slope is [tex]m=-\frac{1}{2}[/tex]
Substituting the slope [tex]m=-\frac{1}{2}[/tex] and the point [tex](8,-8)[/tex] in the formula [tex]y-y_1=m(x-x_1)[/tex], we get,
[tex]y+8=-\frac{1}{2} (x-8)[/tex]
Simplifying, we get,
[tex]y+8=-\frac{1}{2} x+4[/tex]
Subtracting both sides by 8, we get,
[tex]y=-\frac{1}{2} x-4[/tex]
Thus, the equation of the line is [tex]y=-\frac{1}{2} x-4[/tex]
Therefore, Option B is the correct answer.
