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Sagot :
To solve the problem, let's follow these steps:
1. Identify the Equation of the Given Line:
The given equation of the line is [tex]\(5x - 2y = 0\)[/tex]. This line can be rewritten in the slope-intercept form [tex]\(y = mx + b\)[/tex], which is easier for slope identification.
2. Convert to Slope-Intercept Form:
Rearrange the equation [tex]\(5x - 2y = 0\)[/tex] to find the slope.
[tex]\[ 5x - 2y = 0 \implies 2y = 5x \implies y = \frac{5}{2}x \][/tex]
The slope ([tex]\(m\)[/tex]) of this line is [tex]\(\frac{5}{2}\)[/tex].
3. Find the Slope of the Perpendicular Line:
The product of the slopes of two perpendicular lines is [tex]\(-1\)[/tex]. If the slope of the given line is [tex]\(\frac{5}{2}\)[/tex], the slope of the perpendicular line ([tex]\(m_{\perp}\)[/tex]) is:
[tex]\[ \left(\frac{5}{2}\right) \times m_{\perp} = -1 \implies m_{\perp} = -\frac{2}{5} \][/tex]
4. Equation for Line I:
Since line I passes through the origin ([tex]\((0, 0)\)[/tex]), its equation in slope-intercept form [tex]\(y = mx\)[/tex] is:
[tex]\[ y = -\frac{2}{5}x \][/tex]
5. Check the Given Choices:
Now compare this equation with the given multiple choice options to find which one matches:
[tex]\[ \text{1. } 5x - 2y = 8 \][/tex]
This is not in slope-intercept form, and its intercept ([tex]\(b\)[/tex]) is not [tex]\(0\)[/tex], so it's not passing through the origin.
[tex]\[ \text{2. } y = \frac{2}{5}x \][/tex]
The slope here is [tex]\(\frac{2}{5}\)[/tex], which is not the same as [tex]\(-\frac{2}{5}\)[/tex].
[tex]\[ \text{3. } 2x + 5y = 0 \][/tex]
Let's convert this into slope-intercept form:
[tex]\[ 2x + 5y = 0 \implies 5y = -2x \implies y = -\frac{2}{5}x \][/tex]
This matches our equation with a slope of [tex]\(-\frac{2}{5}\)[/tex] and passes through the origin.
[tex]\[ \text{4. } y = -\frac{5}{2}x + 4 \][/tex]
The slope is [tex]\(-\frac{5}{2}\)[/tex], which doesn't match our required slope of [tex]\(-\frac{2}{5}\)[/tex].
Therefore, the correct equation of line I that is perpendicular to [tex]\(5x - 2y = 0\)[/tex] and passes through the origin is:
[tex]\[ \boxed{2x + 5y = 0} \][/tex]
1. Identify the Equation of the Given Line:
The given equation of the line is [tex]\(5x - 2y = 0\)[/tex]. This line can be rewritten in the slope-intercept form [tex]\(y = mx + b\)[/tex], which is easier for slope identification.
2. Convert to Slope-Intercept Form:
Rearrange the equation [tex]\(5x - 2y = 0\)[/tex] to find the slope.
[tex]\[ 5x - 2y = 0 \implies 2y = 5x \implies y = \frac{5}{2}x \][/tex]
The slope ([tex]\(m\)[/tex]) of this line is [tex]\(\frac{5}{2}\)[/tex].
3. Find the Slope of the Perpendicular Line:
The product of the slopes of two perpendicular lines is [tex]\(-1\)[/tex]. If the slope of the given line is [tex]\(\frac{5}{2}\)[/tex], the slope of the perpendicular line ([tex]\(m_{\perp}\)[/tex]) is:
[tex]\[ \left(\frac{5}{2}\right) \times m_{\perp} = -1 \implies m_{\perp} = -\frac{2}{5} \][/tex]
4. Equation for Line I:
Since line I passes through the origin ([tex]\((0, 0)\)[/tex]), its equation in slope-intercept form [tex]\(y = mx\)[/tex] is:
[tex]\[ y = -\frac{2}{5}x \][/tex]
5. Check the Given Choices:
Now compare this equation with the given multiple choice options to find which one matches:
[tex]\[ \text{1. } 5x - 2y = 8 \][/tex]
This is not in slope-intercept form, and its intercept ([tex]\(b\)[/tex]) is not [tex]\(0\)[/tex], so it's not passing through the origin.
[tex]\[ \text{2. } y = \frac{2}{5}x \][/tex]
The slope here is [tex]\(\frac{2}{5}\)[/tex], which is not the same as [tex]\(-\frac{2}{5}\)[/tex].
[tex]\[ \text{3. } 2x + 5y = 0 \][/tex]
Let's convert this into slope-intercept form:
[tex]\[ 2x + 5y = 0 \implies 5y = -2x \implies y = -\frac{2}{5}x \][/tex]
This matches our equation with a slope of [tex]\(-\frac{2}{5}\)[/tex] and passes through the origin.
[tex]\[ \text{4. } y = -\frac{5}{2}x + 4 \][/tex]
The slope is [tex]\(-\frac{5}{2}\)[/tex], which doesn't match our required slope of [tex]\(-\frac{2}{5}\)[/tex].
Therefore, the correct equation of line I that is perpendicular to [tex]\(5x - 2y = 0\)[/tex] and passes through the origin is:
[tex]\[ \boxed{2x + 5y = 0} \][/tex]
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