Question

which is the equation of the line that passes through (6,2) and is perpendicular to a line with slope $-\frac{1}{3}$?
a. $y - 6 = -\frac{1}{3}(x - 2)$
b. $y - 2 = \frac{1}{3}(x - 6)$
c. $y - 2 = 3(x - 6)$
d. $y - 6 = -3(x - 2)$
e. $y - 2 = -3(x - 6)$

Explanation:

Step1: Find the slope of the perpendicular line

The slope of a line perpendicular to a line with slope \( m \) is the negative reciprocal of \( m \). Given the slope of the original line is \( -\frac{1}{3} \), the slope of the perpendicular line \( m_{\perp} \) is calculated as follows:
The negative reciprocal of \( -\frac{1}{3} \) is \( 3 \) (since \( m_{\perp}=-\frac{1}{m}=-\frac{1}{-\frac{1}{3}} = 3 \)).

Step2: Use the point - slope form of a line

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1) \) is a point on the line and \( m \) is the slope of the line.
We know that the line passes through the point \( (6,2) \), so \( x_1 = 6 \) and \( y_1=2 \), and the slope \( m = 3 \) (from Step 1).
Substituting these values into the point - slope form, we get \( y - 2=3(x - 6) \).

Answer:

C. \( y - 2 = 3(x - 6) \)