Find the Equation of the Line Perpendicular to ax + y - 8a = 0 Passing Through (0, 4) (a = 1/2)

To find the equation of the line that passes through point P (0, 4) and is perpendicular to line L, we need to determine the slope of line L and then find the negative reciprocal of this slope to get the slope of the perpendicular line.

Given line L: ax + y - 8a = 0, where a = 1/2, we can rearrange the equation to slope-intercept form (y = mx + b), where m is the slope:

ax + y - 8a = 0 y = -ax + 8a

Comparing this equation with y = mx + b, we see that the slope of line L is -a. Since a = 1/2, the slope of line L is -1/2.

To find the slope of the perpendicular line, we take the negative reciprocal of -1/2: m_perpendicular = -1 / (-1/2) = 2

Now that we have the slope of the perpendicular line, we can use the point-slope form of a line to find the equation. The point-slope form of a line is given by: y - y1 = m(x - x1)

Using point P (0, 4) and the slope m = 2, we can substitute these values into the point-slope form: y - 4 = 2(x - 0)

Simplifying this equation gives: y - 4 = 2x

The equation of the line that passes through P (0, 4) and is perpendicular to line L is y = 2x.