To find an equation of a line that is perpendicular to the given line [tex]\( y - 4 = 2(x - 6) \)[/tex] and passes through the point [tex]\((-3, -5)\)[/tex], let's follow the steps systematically.
1. Convert the given equation to slope-intercept form:
Start with the given equation:
[tex]\[ y - 4 = 2(x - 6) \][/tex]
Distribute the 2 on the right-hand side:
[tex]\[ y - 4 = 2x - 12 \][/tex]
Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 2x - 8 \][/tex]
This is now in the slope-intercept form [tex]\( y = mx + b \)[/tex], where the slope ([tex]\( m \)[/tex]) is 2.
2. Find the slope of the perpendicular line:
Perpendicular lines have slopes that are negative reciprocals of each other. Therefore, the slope of the line perpendicular to the one given will be:
[tex]\[
m = -\frac{1}{2}
\][/tex]
3. Use the point-slope form to find the new equation:
The point-slope form of a line is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\((x_1, y_1) \)[/tex] is a point on the line.
Substitute the point [tex]\((-3, -5) \)[/tex] and the slope [tex]\( -\frac{1}{2} \)[/tex]:
[tex]\[
y - (-5) = -\frac{1}{2}(x - (-3))
\][/tex]
Simplify the equation:
[tex]\[
y + 5 = -\frac{1}{2}(x + 3)
\][/tex]
4. Verification:
The expression [tex]\[ y + 5 = -\frac{1}{2}(x + 3) \][/tex] matches the form of option A.
Therefore, the equation of the line that is perpendicular to [tex]\( y - 4 = 2(x - 6) \)[/tex] and passes through the point [tex]\((-3, -5)\)[/tex] is:
[tex]\[ \boxed{y + 5 = -\frac{1}{2}(x + 3)} \][/tex]
