Question:
If the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ passes through the origin, has radius 3, and its center lies on the line $x + y = 4$, then $g + f =$
If the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ passes through the origin, has radius 3, and its center lies on the line $x + y = 4$, then $g + f =$
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Do not waste time using the radius or origin conditions! The question only requires matching the center $(-g, -f)$ on the line $x+y=4$, which directly yields $g+f = -4$ regardless of other parameters.
Updated On: Jun 3, 2026
- $-4$
- $4$
- $-2$
- $2$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
For the general equation of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$, the center coordinates are $(-g, -f)$.
Step 2: Meaning
Since the center lies on the straight line $x + y = 4$, the coordinates of the center must satisfy the equation of the line.
Step 3: Analysis
Substitute the center $(-g, -f)$ into the line equation $x + y = 4$: \[ (-g) + (-f) = 4 \] \[ -(g + f) = 4 \implies g + f = -4 \]
Step 4: Conclusion
The value of $g + f$ is $-4$.
Final Answer: (A)
For the general equation of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$, the center coordinates are $(-g, -f)$.
Step 2: Meaning
Since the center lies on the straight line $x + y = 4$, the coordinates of the center must satisfy the equation of the line.
Step 3: Analysis
Substitute the center $(-g, -f)$ into the line equation $x + y = 4$: \[ (-g) + (-f) = 4 \] \[ -(g + f) = 4 \implies g + f = -4 \]
Step 4: Conclusion
The value of $g + f$ is $-4$.
Final Answer: (A)
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