Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
Certainly! Let's break down the problem into two parts:
1. Finding the equation of the line through the point (2, -5) and parallel to the given line [tex]\(3x + 4y + 5 = 0\)[/tex].
2. Finding the equation of the line through the same point (2, -5) but perpendicular to the given line.
### Part 1: Line Parallel to [tex]\(3x + 4y + 5 = 0\)[/tex]
1. Find the slope of the given line:
The given line is [tex]\(3x + 4y + 5 = 0\)[/tex]. We need it in the slope-intercept form [tex]\(y = mx + c\)[/tex]:
[tex]\[ 4y = -3x - 5 \][/tex]
[tex]\[ y = -\frac{3}{4}x - \frac{5}{4} \][/tex]
Therefore, the slope of the given line is [tex]\(-\frac{3}{4}\)[/tex].
2. Use the point-slope form for the new line that passes through the point (2, -5) and has the same slope:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where [tex]\(m = -\frac{3}{4}\)[/tex], [tex]\((x_1, y_1) = (2, -5)\)[/tex].
3. Substitute the values:
[tex]\[ y - (-5) = -\frac{3}{4}(x - 2) \][/tex]
Simplifying:
[tex]\[ y + 5 = -\frac{3}{4}(x - 2) \][/tex]
[tex]\[ y + 5 = -\frac{3}{4}x + \frac{3}{2} \][/tex]
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} - 5 \][/tex]
[tex]\[ y = -\frac{3}{4}x - \frac{7}{2} \][/tex]
Converting into the standard form [tex]\(Ax + By + C = 0\)[/tex]:
[tex]\[ 3x + 4y + 14 = 0 \][/tex]
So, the equation of the line parallel to [tex]\(3x + 4y + 5 = 0\)[/tex] and passing through (2, -5) is:
[tex]\[ 3x + 4y + 14 = 0 \][/tex]
### Part 2: Line Perpendicular to [tex]\(3x + 4y + 5 = 0\)[/tex]
1. Slope of the perpendicular line is the negative reciprocal of the original slope [tex]\(-\frac{3}{4}\)[/tex]. So, if the original slope [tex]\(m = -\frac{3}{4}\)[/tex], the slope [tex]\(m_{\perp}\)[/tex] of the perpendicular line will be:
[tex]\[ m_{\perp} = \frac{4}{3} \][/tex]
2. Use the point-slope form for the new line that passes through the point (2, -5) with slope [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[ y - y_1 = m_{\perp}(x - x_1) \][/tex]
Where [tex]\(m_{\perp} = \frac{4}{3}\)[/tex], [tex]\((x_1, y_1) = (2, -5)\)[/tex].
3. Substitute the values:
[tex]\[ y - (-5) = \frac{4}{3}(x - 2) \][/tex]
Simplifying:
[tex]\[ y + 5 = \frac{4}{3}(x - 2) \][/tex]
[tex]\[ y + 5 = \frac{4}{3}x - \frac{8}{3} \][/tex]
[tex]\[ y = \frac{4}{3}x - \frac{8}{3} - 5 \][/tex]
[tex]\[ y = \frac{4}{3}x - \frac{23}{3} \][/tex]
Converting into the standard form [tex]\(Ax + By + C = 0\)[/tex]:
[tex]\[ 4x - 3y - 23 = 0 \][/tex]
So, the equation of the line perpendicular to [tex]\(3x + 4y + 5 = 0\)[/tex] and passing through (2, -5) is:
[tex]\[ 4x - 3y - 23 = 0 \][/tex]
### Conclusion:
- The equation of the line through (2, -5) and parallel to [tex]\(3x + 4y + 5 = 0\)[/tex] is: [tex]\(3x + 4y + 14 = 0\)[/tex].
- The equation of the line through (2, -5) and perpendicular to [tex]\(3x + 4y + 5 = 0\)[/tex] is: [tex]\(4x - 3y - 23 = 0\)[/tex].
1. Finding the equation of the line through the point (2, -5) and parallel to the given line [tex]\(3x + 4y + 5 = 0\)[/tex].
2. Finding the equation of the line through the same point (2, -5) but perpendicular to the given line.
### Part 1: Line Parallel to [tex]\(3x + 4y + 5 = 0\)[/tex]
1. Find the slope of the given line:
The given line is [tex]\(3x + 4y + 5 = 0\)[/tex]. We need it in the slope-intercept form [tex]\(y = mx + c\)[/tex]:
[tex]\[ 4y = -3x - 5 \][/tex]
[tex]\[ y = -\frac{3}{4}x - \frac{5}{4} \][/tex]
Therefore, the slope of the given line is [tex]\(-\frac{3}{4}\)[/tex].
2. Use the point-slope form for the new line that passes through the point (2, -5) and has the same slope:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Where [tex]\(m = -\frac{3}{4}\)[/tex], [tex]\((x_1, y_1) = (2, -5)\)[/tex].
3. Substitute the values:
[tex]\[ y - (-5) = -\frac{3}{4}(x - 2) \][/tex]
Simplifying:
[tex]\[ y + 5 = -\frac{3}{4}(x - 2) \][/tex]
[tex]\[ y + 5 = -\frac{3}{4}x + \frac{3}{2} \][/tex]
[tex]\[ y = -\frac{3}{4}x + \frac{3}{2} - 5 \][/tex]
[tex]\[ y = -\frac{3}{4}x - \frac{7}{2} \][/tex]
Converting into the standard form [tex]\(Ax + By + C = 0\)[/tex]:
[tex]\[ 3x + 4y + 14 = 0 \][/tex]
So, the equation of the line parallel to [tex]\(3x + 4y + 5 = 0\)[/tex] and passing through (2, -5) is:
[tex]\[ 3x + 4y + 14 = 0 \][/tex]
### Part 2: Line Perpendicular to [tex]\(3x + 4y + 5 = 0\)[/tex]
1. Slope of the perpendicular line is the negative reciprocal of the original slope [tex]\(-\frac{3}{4}\)[/tex]. So, if the original slope [tex]\(m = -\frac{3}{4}\)[/tex], the slope [tex]\(m_{\perp}\)[/tex] of the perpendicular line will be:
[tex]\[ m_{\perp} = \frac{4}{3} \][/tex]
2. Use the point-slope form for the new line that passes through the point (2, -5) with slope [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[ y - y_1 = m_{\perp}(x - x_1) \][/tex]
Where [tex]\(m_{\perp} = \frac{4}{3}\)[/tex], [tex]\((x_1, y_1) = (2, -5)\)[/tex].
3. Substitute the values:
[tex]\[ y - (-5) = \frac{4}{3}(x - 2) \][/tex]
Simplifying:
[tex]\[ y + 5 = \frac{4}{3}(x - 2) \][/tex]
[tex]\[ y + 5 = \frac{4}{3}x - \frac{8}{3} \][/tex]
[tex]\[ y = \frac{4}{3}x - \frac{8}{3} - 5 \][/tex]
[tex]\[ y = \frac{4}{3}x - \frac{23}{3} \][/tex]
Converting into the standard form [tex]\(Ax + By + C = 0\)[/tex]:
[tex]\[ 4x - 3y - 23 = 0 \][/tex]
So, the equation of the line perpendicular to [tex]\(3x + 4y + 5 = 0\)[/tex] and passing through (2, -5) is:
[tex]\[ 4x - 3y - 23 = 0 \][/tex]
### Conclusion:
- The equation of the line through (2, -5) and parallel to [tex]\(3x + 4y + 5 = 0\)[/tex] is: [tex]\(3x + 4y + 14 = 0\)[/tex].
- The equation of the line through (2, -5) and perpendicular to [tex]\(3x + 4y + 5 = 0\)[/tex] is: [tex]\(4x - 3y - 23 = 0\)[/tex].
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
