What is the equation of a line that is perpendicular to [tex]y = 4x + 5[/tex] and passes through the point [tex](8, 3)[/tex]?

A. [tex]y = \frac{1}{4} x + 1[/tex]
B. [tex]y = -\frac{1}{4} x - 8[/tex]
C. [tex]y = -\frac{1}{4} x + 5[/tex]
D. [tex]y = -\frac{1}{4} x + 3[/tex]



Answer :

To determine the equation of a line that is perpendicular to the given line \( y = 4x + 5 \) and passes through the point \( (8, 3) \), we follow these steps:

1. Identify the Slope of the Given Line:
- The given line \( y = 4x + 5 \) is in the slope-intercept form \( y = mx + b \) where \( m \) is the slope.
- Here, the slope \( m = 4 \).

2. Calculate the Slope of the Perpendicular Line:
- The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line.
- Therefore, the slope of the perpendicular line is \( -\frac{1}{4} \).

3. Use the Point-Slope Form to Find the Equation:
- We have the slope \( m = -\frac{1}{4} \) and a point \( (8, 3) \) through which the perpendicular line passes.
- The point-slope form of the equation of a line is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line.
- Plug in the values: \( x_1 = 8 \) and \( y_1 = 3 \).

[tex]\[ y - 3 = -\frac{1}{4}(x - 8) \][/tex]

4. Simplify the Equation:
- Distribute the slope on the right-hand side:

[tex]\[ y - 3 = -\frac{1}{4}x + 2 \][/tex]

- Add 3 to both sides to solve for \( y \):

[tex]\[ y = -\frac{1}{4}x + 2 + 3 \][/tex]

- Combine the constants:

[tex]\[ y = -\frac{1}{4}x + 5 \][/tex]

Therefore, the equation of the line that is perpendicular to \( y = 4x + 5 \) and passes through the point \( (8, 3) \) is:

[tex]\[ y = -\frac{1}{4}x + 5 \][/tex]

From the given options, the correct answer is:
C. [tex]\( y = -\frac{1}{4}x + 5 \)[/tex]