Question:
The length of the common chord of the two circles $(x-a)²+(y-b)²=c²$ and $(x-b)²+(y-a)²=c²$ is
The length of the common chord of the two circles $(x-a)²+(y-b)²=c²$ and $(x-b)²+(y-a)²=c²$ is
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The length of the common chord of the two circles $(x-a)=c$ and $(x-b)+(y-a)=c$ is
Updated On: Apr 15, 2026
- $\sqrt{4c^{2}+2(a-b)^{2}}$
- $\sqrt{4c^{2}-2(a-b)^{2}}$
- $\sqrt{4c^{2}-(a-b)^{2}}$
- $\sqrt{2c^{2}-2(a-b)^{2}}$
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The Correct Option is B
Solution and Explanation
Step 1: Concept
The common chord is found by subtracting the equations of the two circles ($S_1 - S_2 = 0$).
Step 2: Analysis
$(x-a)^2 - (x-b)^2 + (y-b)^2 - (y-a)^2 = 0$, which simplifies to the line $x - y = 0$.
Step 3: Evaluation
The distance from centre $(a, b)$ to the chord $x-y=0$ is $M = \frac{|a-b|}{\sqrt{2}}$. Length of chord = $2\sqrt{r^2 - M^2} = 2\sqrt{c^2 - \frac{(a-b)^2}{2}}$.
Step 4: Conclusion
Length $= \sqrt{4c^2 - 2(a-b)^2}$.
Final Answer: (b)
The common chord is found by subtracting the equations of the two circles ($S_1 - S_2 = 0$).
Step 2: Analysis
$(x-a)^2 - (x-b)^2 + (y-b)^2 - (y-a)^2 = 0$, which simplifies to the line $x - y = 0$.
Step 3: Evaluation
The distance from centre $(a, b)$ to the chord $x-y=0$ is $M = \frac{|a-b|}{\sqrt{2}}$. Length of chord = $2\sqrt{r^2 - M^2} = 2\sqrt{c^2 - \frac{(a-b)^2}{2}}$.
Step 4: Conclusion
Length $= \sqrt{4c^2 - 2(a-b)^2}$.
Final Answer: (b)
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