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A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:
- a)3x + 2y= 6
- b)2y + 3x = xy
- c)3x +2 y= xy
- d)3x + 2 y= 6xy
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
A straight line through a fixed point (2, 3) intersects the coordinate...
Let R = (h,k)
P = (0, k)
Q = (h,0)
Equation of line would be,
2k + 3h = hk
Locus of (h, k) is 2y + 3x = xy
P = (0, k)
Q = (h,0)
Equation of line would be,
2k + 3h = hk
Locus of (h, k) is 2y + 3x = xy
Most Upvoted Answer
A straight line through a fixed point (2, 3) intersects the coordinate...
Explanation:
To find the locus of point R, we need to find a relation between the coordinates of point R. Let the coordinates of R be (x, y).
Step 1: Find the equations of lines OP and OQ
Since P lies on the x-axis, the coordinates of P are (x, 0). The equation of line OP passing through (2, 3) and (x, 0) can be found using the slope-intercept form of a straight line:
m = (0 - 3) / (x - 2) = -3 / (x - 2)
Using point-slope form, the equation of line OP is:
y - 3 = -3 / (x - 2) * (x - 2)
Simplifying, we get:
y - 3 = -3
y = 0
Therefore, the equation of line OP is y = 0.
Since Q lies on the y-axis, the coordinates of Q are (0, y). The equation of line OQ passing through (2, 3) and (0, y) can be found using the slope-intercept form of a straight line:
m = (y - 3) / (0 - 2) = (y - 3) / -2
Using point-slope form, the equation of line OQ is:
y - 3 = (y - 3) / -2 * x
Simplifying, we get:
2y - 6 = -x(y - 3)
2y - 6 = -xy + 3x
xy + 2y = 3x + 6
xy + 2y - 3x = 6
2y - 3x + xy = 6
Therefore, the equation of line OQ is 2y - 3x + xy = 6.
Step 2: Find the coordinates of point R
Since R lies on the rectangle OPRQ, it intersects both the x-axis and y-axis. Therefore, the coordinates of R are (x, 0) and (0, y).
Step 3: Find the relation between the coordinates of R
Since R lies on both line OP and line OQ, it satisfies the equations of both lines. Therefore, we can substitute the coordinates of R into both equations to find a relation between the coordinates.
Substituting (x, 0) into the equation of line OP (y = 0), we get:
0 = 0
This equation is always true, so it does not give us any information about the relation between x and y.
Substituting (0, y) into the equation of line OQ (2y - 3x + xy = 6), we get:
2y - 3(0) + 0y = 6
2y = 6
y = 3
Therefore, the y-coordinate of point R is always 3, regardless of the value of x.
Step 4: Find the equation of the locus of R
Since the y-coordinate of point R is always 3, the equation of the locus of R can be written as:
y = 3
Simplifying, we get:
2y -
To find the locus of point R, we need to find a relation between the coordinates of point R. Let the coordinates of R be (x, y).
Step 1: Find the equations of lines OP and OQ
Since P lies on the x-axis, the coordinates of P are (x, 0). The equation of line OP passing through (2, 3) and (x, 0) can be found using the slope-intercept form of a straight line:
m = (0 - 3) / (x - 2) = -3 / (x - 2)
Using point-slope form, the equation of line OP is:
y - 3 = -3 / (x - 2) * (x - 2)
Simplifying, we get:
y - 3 = -3
y = 0
Therefore, the equation of line OP is y = 0.
Since Q lies on the y-axis, the coordinates of Q are (0, y). The equation of line OQ passing through (2, 3) and (0, y) can be found using the slope-intercept form of a straight line:
m = (y - 3) / (0 - 2) = (y - 3) / -2
Using point-slope form, the equation of line OQ is:
y - 3 = (y - 3) / -2 * x
Simplifying, we get:
2y - 6 = -x(y - 3)
2y - 6 = -xy + 3x
xy + 2y = 3x + 6
xy + 2y - 3x = 6
2y - 3x + xy = 6
Therefore, the equation of line OQ is 2y - 3x + xy = 6.
Step 2: Find the coordinates of point R
Since R lies on the rectangle OPRQ, it intersects both the x-axis and y-axis. Therefore, the coordinates of R are (x, 0) and (0, y).
Step 3: Find the relation between the coordinates of R
Since R lies on both line OP and line OQ, it satisfies the equations of both lines. Therefore, we can substitute the coordinates of R into both equations to find a relation between the coordinates.
Substituting (x, 0) into the equation of line OP (y = 0), we get:
0 = 0
This equation is always true, so it does not give us any information about the relation between x and y.
Substituting (0, y) into the equation of line OQ (2y - 3x + xy = 6), we get:
2y - 3(0) + 0y = 6
2y = 6
y = 3
Therefore, the y-coordinate of point R is always 3, regardless of the value of x.
Step 4: Find the equation of the locus of R
Since the y-coordinate of point R is always 3, the equation of the locus of R can be written as:
y = 3
Simplifying, we get:
2y -
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A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:a)3x + 2y= 6b)2y + 3x = xyc)3x +2 y= xyd)3x + 2 y= 6xyCorrect answer is option 'B'. Can you explain this answer?
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A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:a)3x + 2y= 6b)2y + 3x = xyc)3x +2 y= xyd)3x + 2 y= 6xyCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:a)3x + 2y= 6b)2y + 3x = xyc)3x +2 y= xyd)3x + 2 y= 6xyCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:a)3x + 2y= 6b)2y + 3x = xyc)3x +2 y= xyd)3x + 2 y= 6xyCorrect answer is option 'B'. Can you explain this answer?.
A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:a)3x + 2y= 6b)2y + 3x = xyc)3x +2 y= xyd)3x + 2 y= 6xyCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:a)3x + 2y= 6b)2y + 3x = xyc)3x +2 y= xyd)3x + 2 y= 6xyCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:a)3x + 2y= 6b)2y + 3x = xyc)3x +2 y= xyd)3x + 2 y= 6xyCorrect answer is option 'B'. Can you explain this answer?.
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