To determine the equation of the line passing through the point [tex]\((0, 2)\)[/tex] and perpendicular to the line [tex]\(y = \frac{1}{4}x + 5\)[/tex], follow these steps:
1. Identify the slope of the given line:
The equation of the given line is in the form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope. Here, the slope [tex]\(m\)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
2. Determine the slope of the perpendicular line:
The slope of a line perpendicular to another line is the negative reciprocal of the given line's slope. Therefore, the slope of the perpendicular line is:
[tex]\[
-\frac{1}{\frac{1}{4}} = -4
\][/tex]
3. Use the point-slope form of a line equation:
The point-slope form of a line is given by [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. Here, the point is [tex]\((0, 2)\)[/tex], and the slope [tex]\(m\)[/tex] is [tex]\(-4\)[/tex].
4. Substitute the known values into the point-slope form:
Substituting the point [tex]\((0, 2)\)[/tex] and the slope [tex]\(-4\)[/tex] into the point-slope form, we get:
[tex]\[
y - 2 = -4(x - 0)
\][/tex]
5. Simplify the equation:
Simplifying the above equation, we have:
[tex]\[
y - 2 = -4x
\][/tex]
Adding 2 to both sides, we obtain the final equation:
[tex]\[
y = -4x + 2
\][/tex]
Thus, the equation of the line passing through the point [tex]\((0, 2)\)[/tex] and perpendicular to the line [tex]\(y = \frac{1}{4}x + 5\)[/tex] is:
[tex]\[
y = -4x + 2
\][/tex]
