Answer: To find the equation of the line that is perpendicular to the given line and passes through the point (0,9), we need to first find the slope of the given line. We can do this by rearranging the equation into slope-intercept form (y = mx + b):
2x + 12y = -1
12y = -2x - 1
y = (-2/12)x - 1/12
y = (-1/6)x - 1/12
So the slope of the given line is -1/6.
To find the slope of the line perpendicular to the given line, we use the fact that the slopes of perpendicular lines are negative reciprocals of each other. That is:
m1 * m2 = -1
where m1 is the slope of the given line and m2 is the slope of the line we want to find.
So, the slope of the line we want to find is the negative reciprocal of -1/6, which is 6.
Now we have the slope (m = 6) and a point on the line (0,9), so we can use the point-slope form of the equation of a line to find the equation:
y - y1 = m(x - x1)
where x1 = 0 and y1 = 9
y - 9 = 6(x - 0)
y - 9 = 6x
y = 6x + 9
Therefore, the equation of the line that is perpendicular to the given line and passes through the point (0,9) is y = 6x + 9, in slope-intercept form.