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], follow these steps:
1. Convert the given line to slope-intercept form ( [tex]\( y = mx + b \)[/tex] ):
Start with the given equation:
[tex]\[
2x + 12y = -1
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
12y = -2x - 1
\][/tex]
Divide by 12:
[tex]\[
y = -\frac{2}{12}x - \frac{1}{12}
\][/tex]
Simplify the expression:
[tex]\[
y = -\frac{1}{6}x - \frac{1}{12}
\][/tex]
So, the slope [tex]\( m_1 \)[/tex] of the given line is [tex]\( -\frac{1}{6} \)[/tex].
2. Find the slope of the line perpendicular to the given line:
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore:
[tex]\[
m_2 = -\frac{1}{-\frac{1}{6}} = 6
\][/tex]
3. Use the point-slope form of the line equation:
The point-slope form is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] is the point the line passes through. In this case, the point is [tex]\((0, 9)\)[/tex]:
Substitute [tex]\( m_2 = 6 \)[/tex], [tex]\( x_1 = 0 \)[/tex], and [tex]\( y_1 = 9 \)[/tex] into the equation:
[tex]\[
y - 9 = 6(x - 0)
\][/tex]
Simplify the equation:
[tex]\[
y - 9 = 6x
\][/tex]
Solving for [tex]\( y \)[/tex]:
[tex]\[
y = 6x + 9
\][/tex]
So, the equation of the line that is perpendicular to [tex]\( 2x + 12y = -1 \)[/tex] and passes through the point [tex]\((0, 9)\)[/tex] is:
[tex]\[
y = 6x + 9
\][/tex]
Hence, the correct answer is:
[tex]\[
y = 6x + 9
\][/tex]
