To determine the equation of the line that passes through the origin and is parallel to line [tex]\( AB \)[/tex], follow these steps:
1. Calculate the slope of line [tex]\( AB \)[/tex]:
The slope [tex]\( m \)[/tex] between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is calculated as:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Given points [tex]\( A(-3,0) \)[/tex] and [tex]\( B(-6,5) \)[/tex]:
[tex]\[
m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-6 + 3} = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
2. Determine the equation of the line parallel to [tex]\( AB \)[/tex] that passes through the origin:
A line parallel to [tex]\( AB \)[/tex] will have the same slope, [tex]\( m = -\frac{5}{3} \)[/tex]. The equation of a line through the origin (0,0) with slope [tex]\( m \)[/tex] can be written in slope-intercept form as:
[tex]\[
y = mx
\][/tex]
Substituting [tex]\( m \)[/tex] gives:
[tex]\[
y = -\frac{5}{3}x
\][/tex]
3. Rewrite in standard form (Ax + By = C):
Rearranging [tex]\( y = -\frac{5}{3}x \)[/tex] to standard form:
[tex]\[
\frac{5}{3}x + y = 0
\][/tex]
Multiply through by 3 to clear the fraction:
[tex]\[
5x + 3y = 0
\][/tex]
After comparing with the given options, none of those options match exactly. Therefore, it indicates that none of the listed choices correctly matches our derived equation.
Thus, the correct answer is:
[tex]\[
\boxed{\text{None of the above}}
\][/tex]
