Question

Passing through left parenthesis 8 comma negative 5 right parenthesis and perpendicular to the line whose equation is x minus 9 y minus 4 equals 0

what is the equation line

Answer 1

To find the equation of the line that passes through the point \((8, -5)\) and is perpendicular to the line given by the equation \(x - 9y - 4 = 0\), we first need to determine the slope of the given line.

1. **Convert the line equation to slope-intercept form (y = mx + b)**:
\[
x - 9y - 4 = 0 \implies 9y = x - 4 \implies y = \frac{1}{9}x - \frac{4}{9}
\]
The slope (\(m\)) of this line is \(\frac{1}{9}\).

2. **Determine the slope of the perpendicular line**:
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Thus, the slope of our perpendicular line is:
\[
m_{\text{perpendicular}} = -\frac{1}{\frac{1}{9}} = -9
\]

3. **Use the point-slope form of the line equation**:
The point-slope form of a line equation is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope. Using the point \((8, -5)\) and the slope \(-9\):
\[
y - (-5) = -9(x - 8)
\]
Simplifying this:
\[
y + 5 = -9(x - 8)
\]
\[
y + 5 = -9x + 72
\]
\[
y = -9x + 72 - 5
\]
\[
y = -9x + 67
\]

Therefore, the equation of the line that passes through the point \((8, -5)\) and is perpendicular to the line \(x - 9y - 4 = 0\) is:
\[
\boxed{y = -9x + 67}
\]