To find the equation of a line that is perpendicular to a given line and passes through a specific point, let's follow a structured step-by-step approach.
Step 1: Determine the Slope of the Given Line
The given line has the equation [tex]\(2x + 12y = -1\)[/tex]. To determine its slope, we need to rewrite this equation in slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.
Starting from the given equation:
[tex]\[2x + 12y = -1\][/tex]
We solve for [tex]\(y\)[/tex]:
[tex]\[12y = -2x - 1\][/tex]
[tex]\[y = -\frac{2}{12}x - \frac{1}{12}\][/tex]
[tex]\[y = -\frac{1}{6}x - \frac{1}{12}\][/tex]
Here, the slope [tex]\(m\)[/tex] of the given line is [tex]\(-\frac{1}{6}\)[/tex].
Step 2: Determine the Slope of the Perpendicular Line
The slope of a line perpendicular to another is the negative reciprocal of the original slope.
Thus, the slope of the line perpendicular to our given line is:
[tex]\[\text{slope of perpendicular line} = -\frac{1}{-\frac{1}{6}} = 6\][/tex]
Step 3: Use the Point-Slope Form to Find the Equation
The perpendicular line must pass through the point [tex]\((0, 9)\)[/tex]. The point-slope form of a line's equation is [tex]\(y - y_1 = m(x - x_1)\)[/tex], where [tex]\((x_1, y_1\)[/tex]) is a point on the line and [tex]\(m\)[/tex] is the slope.
Given:
- The slope [tex]\(m\)[/tex] = 6
- The point [tex]\((0, 9)\)[/tex]
Substituting these values into the point-slope form:
[tex]\[ y - 9 = 6(x - 0) \][/tex]
[tex]\[ y - 9 = 6x \][/tex]
[tex]\[ y = 6x + 9 \][/tex]
So, 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]
Conclusion
The correct choice from the given options is:
[tex]\[ \boxed{y = 6x + 9} \][/tex]
