To determine the equation of the line that passes through the origin (0,0) and is parallel to line [tex]\(AB\)[/tex], proceed as follows:
### Step 1: Calculate the Slope of Line [tex]\(AB\)[/tex]
The slope of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by the formula:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For line [tex]\(AB\)[/tex] with points [tex]\(A(-3,0)\)[/tex] and [tex]\(B(-6,5)\)[/tex]:
[tex]\[
\text{slope}_{AB} = \frac{5 - 0}{-6 - (-3)} = \frac{5 - 0}{-6 + 3} = \frac{5}{-3} = -\frac{5}{3}
\][/tex]
### Step 2: Equation of a Line Parallel to [tex]\(AB\)[/tex] and Passing through the Origin
The slope of lines that are parallel to each other are equal. Thus, the slope of the line passing through the origin and parallel to line [tex]\(AB\)[/tex] is also [tex]\(-\frac{5}{3}\)[/tex].
The equation of a line with slope [tex]\(m\)[/tex] passing through the origin [tex]\((0,0)\)[/tex] can be expressed in slope-intercept form as:
[tex]\[ y = mx + b \][/tex]
Where [tex]\(b\)[/tex] is the y-intercept. Since the line passes through the origin, [tex]\(b = 0\)[/tex].
Therefore, the equation becomes:
[tex]\[ y = -\frac{5}{3}x \][/tex]
### Step 3: Convert to Standard Form [tex]\(Ax + By = C\)[/tex]
To convert the equation [tex]\(y = -\frac{5}{3}x\)[/tex] to standard form:
[tex]\[
y = -\frac{5}{3}x
\][/tex]
Multiply both sides by 3 to clear the fraction:
[tex]\[
3y = -5x
\][/tex]
Rearrange to standard form [tex]\(Ax + By = C\)[/tex]:
[tex]\[
5x + 3y = 0
\][/tex]
### Step 4: Identify the Correct Answer
The equation of the line passing through the origin and parallel to line [tex]\(AB\)[/tex] is [tex]\(5x + 3y = 0\)[/tex].
Thus, the correct choice is:
[tex]\[ \boxed{5 x + 3 y = 0} \][/tex]
Therefore, the answer is:
[tex]\[ \text{Choice A. } 5x + 3y = 0 \][/tex]
