Question:
The equation of the common chord of the circles $x^2 + y^2 - 4x - 4y = 0$ and $x^2 + y^2 - 6x - 8y + 10 = 0$ is:
The equation of the common chord of the circles $x^2 + y^2 - 4x - 4y = 0$ and $x^2 + y^2 - 6x - 8y + 10 = 0$ is:
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The common chord is always a straight line. Simply subtract the two circle equations directly to eliminate the quadratic terms $x^2$ and $y^2$.
Updated On: Jun 3, 2026
- $x + 2y - 5 = 0$
- $2x + y - 5 = 0$
- $x - 2y + 5 = 0$
- $2x - y + 5 = 0$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
The equation of the common chord of two intersecting circles $S_1 = 0$ and $S_2 = 0$ is represented by the linear equation $S_1 - S_2 = 0$.
Step 2: Meaning
We subtract the second circle equation from the first circle equation to eliminate the second-degree terms and get the line equation.
Step 3: Analysis
Let: \[ S_1 = x^2 + y^2 - 4x - 4y = 0 \] \[ S_2 = x^2 + y^2 - 6x - 8y + 10 = 0 \] Subtracting the equations: \[ S_1 - S_2 = (x^2 + y^2 - 4x - 4y) - (x^2 + y^2 - 6x - 8y + 10) = 0 \] \[ -4x + 6x - 4y + 8y - 10 = 0 \] \[ 2x + 4y - 10 = 0 \] Dividing the entire equation by 2: \[ x + 2y - 5 = 0 \]
Step 4: Conclusion
The equation of the common chord of the circles is $x + 2y - 5 = 0$.
Final Answer: (A)
The equation of the common chord of two intersecting circles $S_1 = 0$ and $S_2 = 0$ is represented by the linear equation $S_1 - S_2 = 0$.
Step 2: Meaning
We subtract the second circle equation from the first circle equation to eliminate the second-degree terms and get the line equation.
Step 3: Analysis
Let: \[ S_1 = x^2 + y^2 - 4x - 4y = 0 \] \[ S_2 = x^2 + y^2 - 6x - 8y + 10 = 0 \] Subtracting the equations: \[ S_1 - S_2 = (x^2 + y^2 - 4x - 4y) - (x^2 + y^2 - 6x - 8y + 10) = 0 \] \[ -4x + 6x - 4y + 8y - 10 = 0 \] \[ 2x + 4y - 10 = 0 \] Dividing the entire equation by 2: \[ x + 2y - 5 = 0 \]
Step 4: Conclusion
The equation of the common chord of the circles is $x + 2y - 5 = 0$.
Final Answer: (A)
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