To determine the equation of the line that passes through the origin and is parallel to the line [tex]\( AB \)[/tex], we start by finding the slope of the line [tex]\( AB \)[/tex]. The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] into this formula, we get:
[tex]\[
m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
So, the slope of line [tex]\( AB \)[/tex] is [tex]\(-\frac{5}{3}\)[/tex].
Since parallel lines share the same slope, a line parallel to [tex]\( AB \)[/tex] will also have a slope of [tex]\(-\frac{5}{3}\)[/tex]. The equation of a line in slope-intercept form [tex]\( y = mx + b \)[/tex], and since this line passes through the origin [tex]\((0, 0)\)[/tex], the y-intercept [tex]\( b \)[/tex] is 0. Thus, the equation becomes:
[tex]\[
y = -\frac{5}{3}x
\][/tex]
To express this equation in standard form, we multiply both sides by 3 to clear the fraction:
[tex]\[
3y = -5x
\][/tex]
Rearranging terms, we write:
[tex]\[
5x - 3y = 0
\][/tex]
This matches option A.
Hence, the equation of the line that passes through the origin and is parallel to line [tex]\( AB \)[/tex] is:
[tex]\[
5x - 3y = 0
\][/tex]
So, the correct option is:
[tex]\[
\boxed{5}
\][/tex]
