Option D: [tex]y=-\frac{1}{2} x+4[/tex] is the equation of line
Explanation:
The equation of the line is perpendicular to the line [tex]y-12=2 x-8[/tex] and that passes through the point [tex](2,3)[/tex]
We need to determine the equation of the line.
First, let us determine the slope of the equation.
Since, the equation is perpendicular to the line [tex]y-12=2 x-8[/tex], the slope is [tex]m=2[/tex]
Since, the lines are perpendicular, we have,
[tex]m=-\frac{1}{2}[/tex]
Now, we shall substitute the slope [tex]m=-\frac{1}{2}[/tex] and the point [tex](2,3)[/tex] in the slope - point formula, we have,
[tex]y-y_1=m(x-x_1)[/tex]
[tex]y-3=-\frac{1}{2} (x-2)[/tex]
[tex]y-3=-\frac{1}{2} x+1[/tex]
Hence, writing this equation in slope - intercept form, we have,
[tex]y=-\frac{1}{2} x+1+3[/tex]
[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 D is the correct answer.
