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The locus of the centres of the circles such that the point (2, 3) is the mid point of the chord
5x + 2y = 16 is
- a)2x – 5y + 11 = 0
- b)2x + 5y – 11 = 0
- c)2x + 5y + 11 = 0
- d)None
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
The locus of the centres of the circles such that the point (2, 3) is ...
Slope of the given chord = −5/2
Slope of the line joining the midpoint on the chord and the centre of the circle = (3+f)/(2+g)
(5/2)[(3+f)/(2+g)] = −1
⇒ 15 + 5f = 4 + 2g
⇒ Locus is 2x − 5y + 11 = 0
Slope of the line joining the midpoint on the chord and the centre of the circle = (3+f)/(2+g)
(5/2)[(3+f)/(2+g)] = −1
⇒ 15 + 5f = 4 + 2g
⇒ Locus is 2x − 5y + 11 = 0
Most Upvoted Answer
The locus of the centres of the circles such that the point (2, 3) is ...
Let locus lie on
Y = mx +C
Slope of a line = -5/2
Hence slope of its perpendicular line = 2/5
Hence
y = 2x/5 +C
5y = 2x + 5C
(2, 3) satisfies this
C = 11/5
Hence
Line will be
5y = 2x + 5*11/5
2x - 5y + 11 = 0
Community Answer
The locus of the centres of the circles such that the point (2, 3) is ...
Understanding the Problem
The problem involves finding the locus of the centers of circles for which the point (2, 3) is the midpoint of a chord defined by the line equation 5x + 2y = 16.
Key Concepts
- Midpoint of a Chord: For a circle, if (x1, y1) is the midpoint of a chord, then the center (h, k) must satisfy the condition of being equidistant from the endpoints of the chord.
- Equation of Chord: The line 5x + 2y = 16 can be rewritten in slope-intercept form to understand its orientation.
Finding the Locus
1. Identifying the Line Equation:
The given line can be rearranged to:
- 2y = -5x + 16
- y = (-5/2)x + 8
2. Using the Midpoint:
The midpoint of a chord is given as (2, 3). For any point (h, k) that represents the center of a circle, the line connecting (h, k) to (2, 3) must be perpendicular to the chord.
3. Perpendicular Slope:
The slope of the line 5x + 2y = 16 is -5/2. The slope of the line perpendicular to this is the negative reciprocal, which is 2/5.
4. Equation of the Locus:
Using point-slope form, the equation of the line passing through (2, 3) with a slope of 2/5 is:
- y - 3 = (2/5)(x - 2)
- Rearranging gives: 2x - 5y + 11 = 0
Conclusion
The locus of the centers of the circles is given by the equation 2x - 5y + 11 = 0, which corresponds to option 'A'. This indicates that all centers of circles with (2, 3) as the midpoint of their chords lie along this line.
The problem involves finding the locus of the centers of circles for which the point (2, 3) is the midpoint of a chord defined by the line equation 5x + 2y = 16.
Key Concepts
- Midpoint of a Chord: For a circle, if (x1, y1) is the midpoint of a chord, then the center (h, k) must satisfy the condition of being equidistant from the endpoints of the chord.
- Equation of Chord: The line 5x + 2y = 16 can be rewritten in slope-intercept form to understand its orientation.
Finding the Locus
1. Identifying the Line Equation:
The given line can be rearranged to:
- 2y = -5x + 16
- y = (-5/2)x + 8
2. Using the Midpoint:
The midpoint of a chord is given as (2, 3). For any point (h, k) that represents the center of a circle, the line connecting (h, k) to (2, 3) must be perpendicular to the chord.
3. Perpendicular Slope:
The slope of the line 5x + 2y = 16 is -5/2. The slope of the line perpendicular to this is the negative reciprocal, which is 2/5.
4. Equation of the Locus:
Using point-slope form, the equation of the line passing through (2, 3) with a slope of 2/5 is:
- y - 3 = (2/5)(x - 2)
- Rearranging gives: 2x - 5y + 11 = 0
Conclusion
The locus of the centers of the circles is given by the equation 2x - 5y + 11 = 0, which corresponds to option 'A'. This indicates that all centers of circles with (2, 3) as the midpoint of their chords lie along this line.
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The locus of the centres of the circles such that the point (2, 3) is the mid point of the chord5x + 2y = 16 isa)2x –5y + 11 = 0b)2x + 5y –11 = 0c)2x + 5y + 11 = 0d)NoneCorrect answer is option 'A'. 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 The locus of the centres of the circles such that the point (2, 3) is the mid point of the chord5x + 2y = 16 isa)2x –5y + 11 = 0b)2x + 5y –11 = 0c)2x + 5y + 11 = 0d)NoneCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The locus of the centres of the circles such that the point (2, 3) is the mid point of the chord5x + 2y = 16 isa)2x –5y + 11 = 0b)2x + 5y –11 = 0c)2x + 5y + 11 = 0d)NoneCorrect answer is option 'A'. Can you explain this answer?.
The locus of the centres of the circles such that the point (2, 3) is the mid point of the chord5x + 2y = 16 isa)2x –5y + 11 = 0b)2x + 5y –11 = 0c)2x + 5y + 11 = 0d)NoneCorrect answer is option 'A'. 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 The locus of the centres of the circles such that the point (2, 3) is the mid point of the chord5x + 2y = 16 isa)2x –5y + 11 = 0b)2x + 5y –11 = 0c)2x + 5y + 11 = 0d)NoneCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The locus of the centres of the circles such that the point (2, 3) is the mid point of the chord5x + 2y = 16 isa)2x –5y + 11 = 0b)2x + 5y –11 = 0c)2x + 5y + 11 = 0d)NoneCorrect answer is option 'A'. Can you explain this answer?.
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