To find the equation of the line that is perpendicular to the given line [tex]\(2x + 12y = -1\)[/tex] and passes through the point [tex]\((0, 9)\)[/tex], we need to follow these steps:
1. Convert the given line equation to slope-intercept form ([tex]\(y = mx + b\)[/tex]):
[tex]\[
2x + 12y = -1
\][/tex]
Solve for [tex]\(y\)[/tex]:
[tex]\[
12y = -2x - 1
\][/tex]
[tex]\[
y = \frac{-2}{12}x - \frac{1}{12}
\][/tex]
Simplify the coefficients:
[tex]\[
y = -\frac{1}{6}x - \frac{1}{12}
\][/tex]
So the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{1}{6}\)[/tex].
2. Find the slope of the line that is perpendicular to the given line:
The slope of a line perpendicular to another is the negative reciprocal of the slope of the original line. Therefore, the perpendicular slope [tex]\(m_{\perp}\)[/tex] is:
[tex]\[
m_{\perp} = -\frac{1}{-\frac{1}{6}} = 6
\][/tex]
3. Use the slope-intercept form of the line equation ( [tex]\(y = mx + b\)[/tex] ) and the given point [tex]\((0, 9)\)[/tex] to find the y-intercept [tex]\(b\)[/tex]:
Substitute the point [tex]\((0, 9)\)[/tex] into the equation with the perpendicular slope:
[tex]\[
y = 6x + b
\][/tex]
Using the point [tex]\((0, 9)\)[/tex]:
[tex]\[
9 = 6 \cdot 0 + b
\][/tex]
[tex]\[
b = 9
\][/tex]
4. Write the final equation of the line:
The equation of the line that is perpendicular to the given line and passes through the point [tex]\((0, 9)\)[/tex] is:
[tex]\[
y = 6x + 9
\][/tex]
Thus, the equation in slope-intercept form of the line that is perpendicular to the given line and passes through the point [tex]\((0,9)\)[/tex] is:
[tex]\[
\boxed{y = 6x + 9}
\][/tex]
