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The co-ordinates of the foot of the perpendicular from (0, 0) upon the line x + y = 2 are
- a)(2, –1)
- b)(–2, 1)
- c)(1, 1)
- d)(1, 2)
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
The co-ordinates of the foot of the perpendicular from (0, 0) upon the...
Let P be the foot of the perpendicular. P lies on a line perpendicular to x + y = 2.
∴ Equation of the line on which P lies is of the form : x – y + k = 0
But this line passes through (0, 0).
∴k = 0
Hence, co-ordinates of P may be obtained by solving x + y = 2 and y = x
∴ x = 1, y = 1
Hence, P ≡ (1, 1)
Most Upvoted Answer
The co-ordinates of the foot of the perpendicular from (0, 0) upon the...
Understanding the Problem
To find the foot of the perpendicular from the point (0, 0) to the line x + y = 2, we need to determine where this perpendicular intersects the line.
Equation of the Line
The line can be rewritten in slope-intercept form:
- y = -x + 2
This indicates that the slope of the line is -1.
Slope of the Perpendicular Line
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line from (0, 0) is:
- Slope = 1
Equation of the Perpendicular Line
Using point-slope form (y - y1 = m(x - x1)), we can write the equation of the line passing through (0, 0) with a slope of 1:
- y = x
Finding the Intersection
To find the foot of the perpendicular, we solve the equations of the two lines:
1. y = x
2. y = -x + 2
Substituting the first equation into the second:
- x = -x + 2
- 2x = 2
- x = 1
Now, substituting x back into y = x:
- y = 1
Thus, the coordinates of the foot of the perpendicular are (1, 1).
Conclusion
The correct answer is option 'C': (1, 1). This point lies on both the original line and the perpendicular line from (0, 0).
To find the foot of the perpendicular from the point (0, 0) to the line x + y = 2, we need to determine where this perpendicular intersects the line.
Equation of the Line
The line can be rewritten in slope-intercept form:
- y = -x + 2
This indicates that the slope of the line is -1.
Slope of the Perpendicular Line
The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line from (0, 0) is:
- Slope = 1
Equation of the Perpendicular Line
Using point-slope form (y - y1 = m(x - x1)), we can write the equation of the line passing through (0, 0) with a slope of 1:
- y = x
Finding the Intersection
To find the foot of the perpendicular, we solve the equations of the two lines:
1. y = x
2. y = -x + 2
Substituting the first equation into the second:
- x = -x + 2
- 2x = 2
- x = 1
Now, substituting x back into y = x:
- y = 1
Thus, the coordinates of the foot of the perpendicular are (1, 1).
Conclusion
The correct answer is option 'C': (1, 1). This point lies on both the original line and the perpendicular line from (0, 0).
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The co-ordinates of the foot of the perpendicular from (0, 0) upon the line x + y = 2 area)(2, –1)b)(–2, 1)c)(1, 1)d)(1, 2)Correct answer is option 'C'. Can you explain this answer?
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