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If g2 + f2 = c, then the equation x2 + y2 + 2gx + 2fy + c = 0 will represent
- a)A circle of radius g
- b)A circle of radius f
- c)A circle of diameter √c
- d)A circle of radius zero
Correct answer is option 'D'. Can you explain this answer?
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If g2+ f2= c, then the equation x2+ y2+ 2gx + 2fy + c = 0will represen...
Understanding the Equation of a Circle
The given equation is:
x² + y² + 2gx + 2fy + c = 0.
This can be rearranged to identify the properties of the circle.
Complete the Square
To convert this equation into a standard form, we can complete the square:
1. Rearranging:
- x² + 2gx + y² + 2fy = -c
2. Completing the square:
- (x + g)² - g² + (y + f)² - f² = -c
3. Simplifying:
- (x + g)² + (y + f)² = g² + f² - c
Analyzing the Circle's Properties
From the completed square form, we see that:
- The center of the circle is at (-g, -f).
- The radius squared (r²) is given by: r² = g² + f² - c.
Condition for Radius
Now, according to the problem statement:
g² + f² = c.
Substituting this into the radius formula gives:
r² = g² + f² - g² - f² = 0.
Thus, the radius r = √0 = 0.
Conclusion
Since the radius is zero, the equation represents a single point rather than a circle with a physical radius.
Therefore, the correct answer is:
Option D: A circle of radius zero.
This represents a degenerate circle, which is effectively a point in the plane.
The given equation is:
x² + y² + 2gx + 2fy + c = 0.
This can be rearranged to identify the properties of the circle.
Complete the Square
To convert this equation into a standard form, we can complete the square:
1. Rearranging:
- x² + 2gx + y² + 2fy = -c
2. Completing the square:
- (x + g)² - g² + (y + f)² - f² = -c
3. Simplifying:
- (x + g)² + (y + f)² = g² + f² - c
Analyzing the Circle's Properties
From the completed square form, we see that:
- The center of the circle is at (-g, -f).
- The radius squared (r²) is given by: r² = g² + f² - c.
Condition for Radius
Now, according to the problem statement:
g² + f² = c.
Substituting this into the radius formula gives:
r² = g² + f² - g² - f² = 0.
Thus, the radius r = √0 = 0.
Conclusion
Since the radius is zero, the equation represents a single point rather than a circle with a physical radius.
Therefore, the correct answer is:
Option D: A circle of radius zero.
This represents a degenerate circle, which is effectively a point in the plane.
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If g2+ f2= c, then the equation x2+ y2+ 2gx + 2fy + c = 0will representa)A circle of radius gb)A circle of radius fc)A circle of diameter √cd)A circle of radius zeroCorrect answer is option 'D'. 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 If g2+ f2= c, then the equation x2+ y2+ 2gx + 2fy + c = 0will representa)A circle of radius gb)A circle of radius fc)A circle of diameter √cd)A circle of radius zeroCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If g2+ f2= c, then the equation x2+ y2+ 2gx + 2fy + c = 0will representa)A circle of radius gb)A circle of radius fc)A circle of diameter √cd)A circle of radius zeroCorrect answer is option 'D'. Can you explain this answer?.
If g2+ f2= c, then the equation x2+ y2+ 2gx + 2fy + c = 0will representa)A circle of radius gb)A circle of radius fc)A circle of diameter √cd)A circle of radius zeroCorrect answer is option 'D'. 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 If g2+ f2= c, then the equation x2+ y2+ 2gx + 2fy + c = 0will representa)A circle of radius gb)A circle of radius fc)A circle of diameter √cd)A circle of radius zeroCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If g2+ f2= c, then the equation x2+ y2+ 2gx + 2fy + c = 0will representa)A circle of radius gb)A circle of radius fc)A circle of diameter √cd)A circle of radius zeroCorrect answer is option 'D'. Can you explain this answer?.
Solutions for If g2+ f2= c, then the equation x2+ y2+ 2gx + 2fy + c = 0will representa)A circle of radius gb)A circle of radius fc)A circle of diameter √cd)A circle of radius zeroCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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