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Find the equation of a line passing through the point of intersection of the lines 2x + y = 8 and x - y = 10 and is perpendicular to 3x + 4y 2012-0.?
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Find the equation of a line passing through the point of intersection ...
Step 1: Find the intersection of the lines
To find the point of intersection of the lines given by:
1. 2x + y = 8
2. x - y = 10
We can solve these equations simultaneously.
- From equation 2, express y in terms of x:
- y = x - 10
- Substitute y into equation 1:
- 2x + (x - 10) = 8
- 3x - 10 = 8
- 3x = 18
- x = 6
- Substitute x back to find y:
- y = 6 - 10 = -4
Thus, the point of intersection is (6, -4).
Step 2: Determine the slope of perpendicular line
The line that is perpendicular to another line has a slope that is the negative reciprocal of the original line's slope.
The equation 3x + 4y = 2012 can be rewritten in slope-intercept form (y = mx + b):
- Rearranging gives: 4y = -3x + 2012
- Therefore, y = (-3/4)x + 503
The slope (m) of this line is -3/4.
Thus, the slope of the line we need to find (perpendicular line) is:
- m_perpendicular = 4/3
Step 3: Formulate the equation of the new line
Using the point-slope form of the equation of a line (y - y1 = m(x - x1)):
- Point (x1, y1) = (6, -4)
- Slope (m) = 4/3
The equation becomes:
- y - (-4) = (4/3)(x - 6)
This simplifies to:
- y + 4 = (4/3)x - 8
- y = (4/3)x - 12
Final Equation
The equation of the line passing through the intersection point and perpendicular to the given line is:
- y = (4/3)x - 12.
To find the point of intersection of the lines given by:
1. 2x + y = 8
2. x - y = 10
We can solve these equations simultaneously.
- From equation 2, express y in terms of x:
- y = x - 10
- Substitute y into equation 1:
- 2x + (x - 10) = 8
- 3x - 10 = 8
- 3x = 18
- x = 6
- Substitute x back to find y:
- y = 6 - 10 = -4
Thus, the point of intersection is (6, -4).
Step 2: Determine the slope of perpendicular line
The line that is perpendicular to another line has a slope that is the negative reciprocal of the original line's slope.
The equation 3x + 4y = 2012 can be rewritten in slope-intercept form (y = mx + b):
- Rearranging gives: 4y = -3x + 2012
- Therefore, y = (-3/4)x + 503
The slope (m) of this line is -3/4.
Thus, the slope of the line we need to find (perpendicular line) is:
- m_perpendicular = 4/3
Step 3: Formulate the equation of the new line
Using the point-slope form of the equation of a line (y - y1 = m(x - x1)):
- Point (x1, y1) = (6, -4)
- Slope (m) = 4/3
The equation becomes:
- y - (-4) = (4/3)(x - 6)
This simplifies to:
- y + 4 = (4/3)x - 8
- y = (4/3)x - 12
Final Equation
The equation of the line passing through the intersection point and perpendicular to the given line is:
- y = (4/3)x - 12.
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Find the equation of a line passing through the point of intersection of the lines 2x + y = 8 and x - y = 10 and is perpendicular to 3x + 4y 2012-0.?
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Find the equation of a line passing through the point of intersection of the lines 2x + y = 8 and x - y = 10 and is perpendicular to 3x + 4y 2012-0.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Find the equation of a line passing through the point of intersection of the lines 2x + y = 8 and x - y = 10 and is perpendicular to 3x + 4y 2012-0.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the equation of a line passing through the point of intersection of the lines 2x + y = 8 and x - y = 10 and is perpendicular to 3x + 4y 2012-0.?.
Find the equation of a line passing through the point of intersection of the lines 2x + y = 8 and x - y = 10 and is perpendicular to 3x + 4y 2012-0.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Find the equation of a line passing through the point of intersection of the lines 2x + y = 8 and x - y = 10 and is perpendicular to 3x + 4y 2012-0.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the equation of a line passing through the point of intersection of the lines 2x + y = 8 and x - y = 10 and is perpendicular to 3x + 4y 2012-0.?.
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