To determine the equation of line [tex]\( l \)[/tex] that is perpendicular to the given line [tex]\( 5x - 2y = 8 \)[/tex] and passes through the origin, we need to follow these steps:
### Step 1: Find the Slope of the Given Line
The equation of the given line is [tex]\( 5x - 2y = 8 \)[/tex]. We need to convert this into the slope-intercept form, [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
1. Start with the given equation:
[tex]\[
5x - 2y = 8
\][/tex]
2. Solve for [tex]\( y \)[/tex]:
[tex]\[
-2y = -5x + 8
\][/tex]
[tex]\[
y = \frac{5}{2}x - 4
\][/tex]
From this equation, we see that the slope [tex]\( m \)[/tex] of the given line is [tex]\( \frac{5}{2} \)[/tex].
### Step 2: Determine the Slope of the Perpendicular Line
The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. The negative reciprocal of [tex]\( \frac{5}{2} \)[/tex] is:
[tex]\[
-\frac{2}{5}
\][/tex]
### Step 3: Write the Equation of the Perpendicular Line Through the Origin
Since line [tex]\( l \)[/tex] passes through the origin, its y-intercept [tex]\( b \)[/tex] is 0. Hence, the equation of the line [tex]\( l \)[/tex] with slope [tex]\(-\frac{2}{5}\)[/tex] is:
[tex]\[
y = -\frac{2}{5}x
\][/tex]
### Step 4: Check the Given Options
We now compare the equation [tex]\( y = -\frac{2}{5}x \)[/tex] with the given options to see which one matches.
1. Option 1: [tex]\( 5x - 2y = 8 \)[/tex]
- This is the equation of the original line, so it is not the line [tex]\( l \)[/tex].
2. Option 2: [tex]\( y = \frac{2}{5}x \)[/tex]
- This line has a slope of [tex]\( \frac{2}{5} \)[/tex], which is not the correct negative reciprocal.
3. Option 3: [tex]\( 2x + 5y = 0 \)[/tex]
- To check if this is correct, convert it to the slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
5y = -2x
\][/tex]
[tex]\[
y = -\frac{2}{5}x
\][/tex]
- This is exactly the equation we derived for line [tex]\( l \)[/tex].
4. Option 4: [tex]\( y = -\frac{5}{2}x + 4 \)[/tex]
- This line has a slope of [tex]\(-\frac{5}{2}\)[/tex], which is not the correct slope.
### Conclusion
Based on the calculations, the correct equation of line [tex]\( l \)[/tex] is given by Option 3:
\[
2x + 5y = 0
