To find 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], follow these steps:
1. Identify the slope of the given line:
The given line equation is [tex]\( y - 4 = 2(x - 6) \)[/tex]. This equation is in the point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope.
From the equation, we can identify that the slope [tex]\( m \)[/tex] of the given line is [tex]\( 2 \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of the line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, if the slope of the given line is [tex]\( 2 \)[/tex], the slope of the perpendicular line will be:
[tex]\[
\text{Perpendicular slope} = -\frac{1}{2}
\][/tex]
3. Use the point-slope form to write the equation of the perpendicular line:
The point-slope form of the equation of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\( m \)[/tex] is the slope of the perpendicular line ([tex]\( -\frac{1}{2} \)[/tex]), and [tex]\( (x_1, y_1) \)[/tex] is the point [tex]\((-3, -5)\)[/tex].
4. Substitute the known values:
[tex]\[
y - (-5) = -\frac{1}{2}(x - (-3))
\][/tex]
Simplify the equation:
[tex]\[
y + 5 = -\frac{1}{2}(x + 3)
\][/tex]
5. Compare with the given options:
The derived equation matches option A:
[tex]\[
y + 5 = -\frac{1}{2}(x + 3)
\][/tex]
Thus, the correct 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]
