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Sagot :
To determine the equations that represent the line perpendicular to [tex]\(5x - 2y = -6\)[/tex] and passing through the point [tex]\((5, -4)\)[/tex], follow these steps:
1. Find the slope of the given line:
- First, express [tex]\(5x - 2y = -6\)[/tex] in slope-intercept form [tex]\(y = mx + b\)[/tex].
- Rearrange the equation:
[tex]\[ 5x - 2y = -6 \implies -2y = -5x - 6 \implies y = \frac{5}{2}x + 3 \][/tex]
- From the slope-intercept form, the slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{5}{2}\)[/tex].
2. Determine the slope of the perpendicular line:
- The slope of a line perpendicular to another is the negative reciprocal of the original slope.
- The reciprocal of [tex]\(\frac{5}{2}\)[/tex] is [tex]\(\frac{2}{5}\)[/tex], and taking the negative gives us [tex]\(-\frac{2}{5}\)[/tex].
3. Find the equation of the perpendicular line passing through [tex]\((5, -4)\)[/tex]:
- Use the point-slope form of a line’s equation: [tex]\(y - y_1 = m(x - x1)\)[/tex], where [tex]\((x1, y1)\)[/tex] is the point [tex]\((5, -4)\)[/tex] and [tex]\(m = -\frac{2}{5}\)[/tex].
- Substituting the values, we get:
[tex]\[ y - (-4) = -\frac{2}{5}(x - 5) \][/tex]
- Simplify to get:
[tex]\[ y + 4 = -\frac{2}{5}(x - 5) \][/tex]
4. Convert the equation to standard form [tex]\(Ax + By = C\)[/tex]:
- Multiplying both sides by 5 to eliminate the fraction, we get:
[tex]\[ 5(y + 4) = -2(x - 5) \implies 5y + 20 = -2x + 10 \][/tex]
- Rearrange to standard form:
[tex]\[ 2x + 5y = -10 \][/tex]
5. Identify the correct options:
- From the above steps, verify each given option:
- [tex]\(y = -\frac{2}{5}x - 2\)[/tex]: This is not in point-slope form for the given point.
- [tex]\(2x + 5y = -10\)[/tex]: This matches with the standard form derived.
- [tex]\(2x - 5y = -10\)[/tex]: This does not match the derived form.
- [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]: This matches the point-slope form before conversion.
- [tex]\(y - 4 = \frac{5}{2}(x + 5)\)[/tex]: This matches the inverse case and is irrelevant.
By verifying, we find that the correct equations representing the perpendicular line are:
[tex]\[ \boxed{2, 4, 5} \][/tex]
Thus, the correct options are:
1. [tex]\(2x + 5y = -10\)[/tex]
2. [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]
3. [tex]\(y - 4 = \frac{5}{2}(x + 5)\)[/tex]
1. Find the slope of the given line:
- First, express [tex]\(5x - 2y = -6\)[/tex] in slope-intercept form [tex]\(y = mx + b\)[/tex].
- Rearrange the equation:
[tex]\[ 5x - 2y = -6 \implies -2y = -5x - 6 \implies y = \frac{5}{2}x + 3 \][/tex]
- From the slope-intercept form, the slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{5}{2}\)[/tex].
2. Determine the slope of the perpendicular line:
- The slope of a line perpendicular to another is the negative reciprocal of the original slope.
- The reciprocal of [tex]\(\frac{5}{2}\)[/tex] is [tex]\(\frac{2}{5}\)[/tex], and taking the negative gives us [tex]\(-\frac{2}{5}\)[/tex].
3. Find the equation of the perpendicular line passing through [tex]\((5, -4)\)[/tex]:
- Use the point-slope form of a line’s equation: [tex]\(y - y_1 = m(x - x1)\)[/tex], where [tex]\((x1, y1)\)[/tex] is the point [tex]\((5, -4)\)[/tex] and [tex]\(m = -\frac{2}{5}\)[/tex].
- Substituting the values, we get:
[tex]\[ y - (-4) = -\frac{2}{5}(x - 5) \][/tex]
- Simplify to get:
[tex]\[ y + 4 = -\frac{2}{5}(x - 5) \][/tex]
4. Convert the equation to standard form [tex]\(Ax + By = C\)[/tex]:
- Multiplying both sides by 5 to eliminate the fraction, we get:
[tex]\[ 5(y + 4) = -2(x - 5) \implies 5y + 20 = -2x + 10 \][/tex]
- Rearrange to standard form:
[tex]\[ 2x + 5y = -10 \][/tex]
5. Identify the correct options:
- From the above steps, verify each given option:
- [tex]\(y = -\frac{2}{5}x - 2\)[/tex]: This is not in point-slope form for the given point.
- [tex]\(2x + 5y = -10\)[/tex]: This matches with the standard form derived.
- [tex]\(2x - 5y = -10\)[/tex]: This does not match the derived form.
- [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]: This matches the point-slope form before conversion.
- [tex]\(y - 4 = \frac{5}{2}(x + 5)\)[/tex]: This matches the inverse case and is irrelevant.
By verifying, we find that the correct equations representing the perpendicular line are:
[tex]\[ \boxed{2, 4, 5} \][/tex]
Thus, the correct options are:
1. [tex]\(2x + 5y = -10\)[/tex]
2. [tex]\(y + 4 = -\frac{2}{5}(x - 5)\)[/tex]
3. [tex]\(y - 4 = \frac{5}{2}(x + 5)\)[/tex]
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