Sure, let's solve the problem step-by-step:
We are given the line equation [tex]\( y - 4 = 2(x - 6) \)[/tex] and need to find the equation of a line that is perpendicular to this line and passes through the point [tex]\((-3, -5)\)[/tex].
### Step 1: Identify the slope of the given line
First, we need to convert the given line equation to the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
Given equation:
[tex]\[ y - 4 = 2(x - 6) \][/tex]
Distribute and simplify:
[tex]\[ y - 4 = 2x - 12 \][/tex]
Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 2x - 12 + 4 \][/tex]
[tex]\[ y = 2x - 8 \][/tex]
The slope [tex]\( m \)[/tex] of the given line is 2.
### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope.
The original slope is 2, so the negative reciprocal is:
[tex]\[ -\frac{1}{2} \][/tex]
### Step 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:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where [tex]\( m \)[/tex] is the slope, and [tex]\((x_1, y_1)\)[/tex] is the point through which the line passes.
Plugging in the slope [tex]\( m = -\frac{1}{2} \)[/tex] and the point (-3, -5):
[tex]\[ y - (-5) = -\frac{1}{2}(x - (-3)) \][/tex]
[tex]\[ y + 5 = -\frac{1}{2}(x + 3) \][/tex]
### Step 4: Verify the answer
Among the given options, the equation we derived matches option C:
[tex]\[ y + 5 = -\frac{1}{2}(x + 3) \][/tex]
Therefore, the correct answer is:
C. [tex]\( y + 5 = -\frac{1}{2}(x + 3) \)[/tex]
