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Sagot :
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]
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]
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