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The locus of the mid -point of a chord of the circle x2 + y2 = 4 which subtends a right angle at the origin is (1984 - 2 Marks)
- a)x + y = 2
- b)x2 + y2 = 1
- c)x2 + y2 = 2
- d)x + y = 1
Correct answer is option 'C'. Can you explain this answer?
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The locus of the mid -point of a chord of the circle x2 + y2 = 4 which...
Locus of Mid-point of a Chord Subtending Right Angle at the Origin
Given equation of the circle: x^2 + y^2 = 4
To find the locus of the mid-point of a chord that subtends a right angle at the origin, we need to consider the properties of chords that form right angles with the origin.
1. Chords Subtending Right Angle:
A chord subtends a right angle at the origin if the product of the coordinates of its endpoints is equal to zero. This can be expressed as:
(x1 * x2) + (y1 * y2) = 0
2. Mid-point of a Chord:
The mid-point of a chord can be found by averaging the coordinates of its endpoints. Let (h, k) be the coordinates of the mid-point.
3. Applying the Conditions:
Using the above conditions, we can substitute the coordinates of the mid-point into the equation for chords subtending a right angle at the origin.
(x1 * x2) + (y1 * y2) = 0
((h - 2) * (h + 2)) + (k * k) = 0
(h^2 - 4) + (k^2) = 0
h^2 + k^2 = 4
4. Simplification:
The equation obtained in step 3 is the equation of a circle with center at the origin and radius 2.
5. Conclusion:
Therefore, the locus of the mid-point of a chord of the circle x^2 + y^2 = 4 that subtends a right angle at the origin is the circle with equation:
x^2 + y^2 = 4
Hence, the correct answer is option 'C': x^2 + y^2 = 2.
Given equation of the circle: x^2 + y^2 = 4
To find the locus of the mid-point of a chord that subtends a right angle at the origin, we need to consider the properties of chords that form right angles with the origin.
1. Chords Subtending Right Angle:
A chord subtends a right angle at the origin if the product of the coordinates of its endpoints is equal to zero. This can be expressed as:
(x1 * x2) + (y1 * y2) = 0
2. Mid-point of a Chord:
The mid-point of a chord can be found by averaging the coordinates of its endpoints. Let (h, k) be the coordinates of the mid-point.
3. Applying the Conditions:
Using the above conditions, we can substitute the coordinates of the mid-point into the equation for chords subtending a right angle at the origin.
(x1 * x2) + (y1 * y2) = 0
((h - 2) * (h + 2)) + (k * k) = 0
(h^2 - 4) + (k^2) = 0
h^2 + k^2 = 4
4. Simplification:
The equation obtained in step 3 is the equation of a circle with center at the origin and radius 2.
5. Conclusion:
Therefore, the locus of the mid-point of a chord of the circle x^2 + y^2 = 4 that subtends a right angle at the origin is the circle with equation:
x^2 + y^2 = 4
Hence, the correct answer is option 'C': x^2 + y^2 = 2.
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The locus of the mid -point of a chord of the circle x2 + y2 = 4 which subtends a right angle at the origin is (1984 - 2 Marks)a)x + y = 2b)x2 + y2 = 1c)x2 + y2 = 2d)x + y = 1Correct answer is option 'C'. 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 mid -point of a chord of the circle x2 + y2 = 4 which subtends a right angle at the origin is (1984 - 2 Marks)a)x + y = 2b)x2 + y2 = 1c)x2 + y2 = 2d)x + y = 1Correct answer is option 'C'. 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 mid -point of a chord of the circle x2 + y2 = 4 which subtends a right angle at the origin is (1984 - 2 Marks)a)x + y = 2b)x2 + y2 = 1c)x2 + y2 = 2d)x + y = 1Correct answer is option 'C'. Can you explain this answer?.
The locus of the mid -point of a chord of the circle x2 + y2 = 4 which subtends a right angle at the origin is (1984 - 2 Marks)a)x + y = 2b)x2 + y2 = 1c)x2 + y2 = 2d)x + y = 1Correct answer is option 'C'. 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 mid -point of a chord of the circle x2 + y2 = 4 which subtends a right angle at the origin is (1984 - 2 Marks)a)x + y = 2b)x2 + y2 = 1c)x2 + y2 = 2d)x + y = 1Correct answer is option 'C'. 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 mid -point of a chord of the circle x2 + y2 = 4 which subtends a right angle at the origin is (1984 - 2 Marks)a)x + y = 2b)x2 + y2 = 1c)x2 + y2 = 2d)x + y = 1Correct answer is option 'C'. Can you explain this answer?.
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