To determine the equation of the line passing through the origin (0,0) and parallel to line \(AB\), we follow these steps:
1. Find the Slope of Line \(AB\):
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Here, the coordinates of points \(A\) and \(B\) are:
[tex]\[
A(-3, 0) \quad \text{and} \quad B(-6, 5)
\][/tex]
Substituting these coordinates into the slope formula gives:
[tex]\[
m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-6 + 3} = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
2. Equation of the Line Parallel to \(AB\) Passing through the Origin:
Since the required line is parallel to line \(AB\), it will have the same slope \(-\frac{5}{3}\).
The general form of the equation of a line with slope \(m\) passing through the origin \((0,0)\) is:
[tex]\[
y = mx
\][/tex]
Substituting \(m = -\frac{5}{3}\) into this equation gives:
[tex]\[
y = -\frac{5}{3} x
\][/tex]
3. Convert to Standard Form \(ax + by + c = 0\):
To convert \(y = -\frac{5}{3} x\) into standard form, we can remove the fraction by multiplying all terms by 3:
[tex]\[
3y = -5x
\][/tex]
Rearranging the equation to the standard form \(ax + by + c = 0\) results in:
[tex]\[
5x + 3y = 0
\][/tex]
4. Matching the Standard Form:
The equation is \(5x - 3y = 0\), which can be written as:
[tex]\[
5x - 3y = 0
\][/tex]
This matches option A.
Therefore, the correct answer is:
[tex]\[
\boxed{5x - 3y = 0}
\][/tex]
