Answer :
To find the equation of the line passing through the origin and parallel to the line \(AB\) where \(A(-3,0)\) and \(B(-6,5)\), we need to follow these steps:
1. Determine the Slope of Line \(AB\):
The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points \(A(-3,0)\) and \(B(-6,5)\):
[tex]\[ m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-3} = -\frac{5}{3} \][/tex]
2. Equation of a Line Parallel to \(AB\) Through the Origin:
Since parallel lines have the same slope, the slope of the line through the origin (which we will call L) is also \(-\frac{5}{3}\).
The equation of a line with slope \(m\) passing through the origin \((0, 0)\) is given by \(y = mx\):
[tex]\[ y = -\frac{5}{3}x \][/tex]
3. Convert the Equation to Standard Form:
To present the equation in the form \(Ax + By = 0\), we start with:
[tex]\[ y + \frac{5}{3}x = 0 \][/tex]
To clear the fraction, multiply every term by \(3\):
[tex]\[ 3y + 5x = 0 \][/tex]
4. Rearrange to Match Standard Form:
[tex]\[ 5x + 3y = 0 \][/tex]
Given this form \(5x + 3y = 0\), the correct answer to the equation of the line that passes through the origin and is parallel to line \(AB\) is:
A. [tex]\(5x + 3y = 0\)[/tex]
1. Determine the Slope of Line \(AB\):
The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the points \(A(-3,0)\) and \(B(-6,5)\):
[tex]\[ m = \frac{5 - 0}{-6 - (-3)} = \frac{5}{-3} = -\frac{5}{3} \][/tex]
2. Equation of a Line Parallel to \(AB\) Through the Origin:
Since parallel lines have the same slope, the slope of the line through the origin (which we will call L) is also \(-\frac{5}{3}\).
The equation of a line with slope \(m\) passing through the origin \((0, 0)\) is given by \(y = mx\):
[tex]\[ y = -\frac{5}{3}x \][/tex]
3. Convert the Equation to Standard Form:
To present the equation in the form \(Ax + By = 0\), we start with:
[tex]\[ y + \frac{5}{3}x = 0 \][/tex]
To clear the fraction, multiply every term by \(3\):
[tex]\[ 3y + 5x = 0 \][/tex]
4. Rearrange to Match Standard Form:
[tex]\[ 5x + 3y = 0 \][/tex]
Given this form \(5x + 3y = 0\), the correct answer to the equation of the line that passes through the origin and is parallel to line \(AB\) is:
A. [tex]\(5x + 3y = 0\)[/tex]
