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The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference is
- a)(x - 2)2 + (y + 3)2 = 6.25
- b)(x + 2)2 + (y - 3)2 = 6.25
- c)(x + 2)2 + (y - 3)2 = 18.75
- d)(x + 2)2 + (y + 3)2 = 18.75
Correct answer is option 'B'. Can you explain this answer?
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The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = ...
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The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = ...
To find the locus of the mid-points of the chords that subtend an angle of 90 degrees, we need to find the equation of the circle passing through the endpoints of these chords.
First, let's rewrite the given equation in the standard form of a circle:
x^2 + y^2 + 4x - 6y - 12 = 0
Completing the square for x and y terms, we have:
(x^2 + 4x) + (y^2 - 6y) - 12 = 0
(x^2 + 4x + 4) + (y^2 - 6y + 9) - 12 = 4 + 9
(x + 2)^2 + (y - 3)^2 = 17
So the given equation represents a circle centered at (-2, 3) with a radius of √17.
Now, let's find the endpoints of the chords that subtend a 90-degree angle. Since the angle is right, the chords will be diameters of the circle.
The diameter of a circle passes through its center, so the endpoints of the diameter will be (-2, 3) and another point on the circle.
To find another point on the circle, we can use the equation of the circle and choose a value for x that satisfies it. Let's choose x = 0:
(x + 2)^2 + (y - 3)^2 = 17
(0 + 2)^2 + (y - 3)^2 = 17
4 + (y - 3)^2 = 17
(y - 3)^2 = 17 - 4
(y - 3)^2 = 13
y - 3 = ±√13
y = 3 ± √13
The two points on the circle are (0, 3 + √13) and (0, 3 - √13). The midpoint of these two points will be the center of the chord, which is also the center of the circle (-2, 3).
Therefore, the locus of the mid-points of the chords that subtend an angle of 90 degrees is the point (-2, 3).
First, let's rewrite the given equation in the standard form of a circle:
x^2 + y^2 + 4x - 6y - 12 = 0
Completing the square for x and y terms, we have:
(x^2 + 4x) + (y^2 - 6y) - 12 = 0
(x^2 + 4x + 4) + (y^2 - 6y + 9) - 12 = 4 + 9
(x + 2)^2 + (y - 3)^2 = 17
So the given equation represents a circle centered at (-2, 3) with a radius of √17.
Now, let's find the endpoints of the chords that subtend a 90-degree angle. Since the angle is right, the chords will be diameters of the circle.
The diameter of a circle passes through its center, so the endpoints of the diameter will be (-2, 3) and another point on the circle.
To find another point on the circle, we can use the equation of the circle and choose a value for x that satisfies it. Let's choose x = 0:
(x + 2)^2 + (y - 3)^2 = 17
(0 + 2)^2 + (y - 3)^2 = 17
4 + (y - 3)^2 = 17
(y - 3)^2 = 17 - 4
(y - 3)^2 = 13
y - 3 = ±√13
y = 3 ± √13
The two points on the circle are (0, 3 + √13) and (0, 3 - √13). The midpoint of these two points will be the center of the chord, which is also the center of the circle (-2, 3).
Therefore, the locus of the mid-points of the chords that subtend an angle of 90 degrees is the point (-2, 3).
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The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct 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 The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct 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 The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct answer is option 'B'. Can you explain this answer?.
The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct 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 The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct 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 The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct answer is option 'B'. Can you explain this answer?.
Solutions for The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct answer is option 'B'. Can you explain this answer?, a detailed solution for The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of The locus of the mid-points of the chords of x2 + y2 + 4x - 6y - 12 = 0 which subtend an angle of π 3 at its circumference isa)(x - 2)2 + (y + 3)2 = 6.25b)(x + 2)2 + (y - 3)2 = 6.25c)(x + 2)2 + (y - 3)2 = 18.75d)(x + 2)2 + (y + 3)2 = 18.75Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
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