To determine the equation of the line that is perpendicular to [tex]\( y - 4 = \frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex], follow these steps:
1. Find the slope of the given line:
The equation given is in the point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex], where [tex]\( m \)[/tex] is the slope. The equation is [tex]\( y - 4 = \frac{2}{3}(x - 6) \)[/tex].
So, the slope ([tex]\( m_1 \)[/tex]) of the given line is [tex]\( \frac{2}{3} \)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. If the slope of the given line is [tex]\( \frac{2}{3} \)[/tex], then the slope of the line perpendicular to it is:
[tex]\[
m_2 = -\frac{1}{(\frac{2}{3})} = -\frac{3}{2}
\][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
We use the point [tex]\((-2, -2)\)[/tex] and the slope [tex]\(-\frac{3}{2}\)[/tex] and substitute them into the point-slope form equation, which is [tex]\( y - y_1 = m(x - x_1) \)[/tex].
[tex]\[
y - (-2) = -\frac{3}{2}(x - (-2))
\][/tex]
Simplify the equation:
[tex]\[
y + 2 = -\frac{3}{2}(x + 2)
\][/tex]
4. Convert this into slope-intercept form ([tex]\( y = mx + b \)[/tex]):
Distribute the slope [tex]\(-\frac{3}{2}\)[/tex] across the terms inside the parentheses:
[tex]\[
y + 2 = -\frac{3}{2}x - \frac{3}{2} \cdot 2
\][/tex]
[tex]\[
y + 2 = -\frac{3}{2}x - 3
\][/tex]
Now, isolate [tex]\( y \)[/tex] by subtracting 2 from both sides:
[tex]\[
y = -\frac{3}{2}x - 3 - 2
\][/tex]
[tex]\[
y = -\frac{3}{2}x - 5
\][/tex]
The equation of the line that is perpendicular to [tex]\( y - 4 = \frac{2}{3}(x - 6) \)[/tex] and passes through the point [tex]\((-2, -2)\)[/tex] is:
[tex]\[
y = -\frac{3}{2}x - 5
\][/tex]
However, this exact slope-intercept form of the equation isn’t presented in the provided choices. Given the options, none of them are correct because the correct equation for the perpendicular line should be:
[tex]\[
y = -\frac{3}{2}x - 5
\][/tex]
Thus, we can conclude that none of the options provided match the correct equation.
