What is the equation of the line that passes through the origin and is parallel to the line [tex] AB [/tex], which passes through [tex] A(-3,0) [/tex] and [tex] B(-6,5) [/tex]?

A. [tex] 5x - 3y = 0 [/tex]
B. [tex] -x + 3y = 0 [/tex]
C. [tex] -5x - 3y = 0 [/tex]
D. [tex] 3x + 5y = 0 [/tex]
E. [tex] -3x + 5y = 0 [/tex]



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]