Question:
If the circle $x^2 + y^2 + 2x - 2y + c = 0$ cuts the circle $x^2 + y^2 - 4x - 6y + 11 = 0$ orthogonally, then the value of $c$ is:
If the circle $x^2 + y^2 + 2x - 2y + c = 0$ cuts the circle $x^2 + y^2 - 4x - 6y + 11 = 0$ orthogonally, then the value of $c$ is:
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Memorize the orthogonality condition: $2g_1g_2 + 2f_1f_2 = c_1 + c_2$. Be careful to keep the correct signs of $g$ and $f$.
Updated On: Jun 3, 2026
- $-9$
- $9$
- $-13$
- $13$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
Two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ cut each other orthogonally if they satisfy the condition: \[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \]
Step 2: Meaning
We identify the parameters of both circles: First circle: $g_1 = 1$, $f_1 = -1$, $c_1 = c$ Second circle: $g_2 = -2$, $f_2 = -3$, $c_2 = 11$
Step 3: Analysis
Substitute the parameters into the orthogonality condition: \[ 2(1)(-2) + 2(-1)(-3) = c + 11 \] \[ -4 + 6 = c + 11 \] \[ 2 = c + 11 \implies c = -9 \]
Step 4: Conclusion
The value of $c$ is $-9$.
Final Answer: (A)
Two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ cut each other orthogonally if they satisfy the condition: \[ 2g_1g_2 + 2f_1f_2 = c_1 + c_2 \]
Step 2: Meaning
We identify the parameters of both circles: First circle: $g_1 = 1$, $f_1 = -1$, $c_1 = c$ Second circle: $g_2 = -2$, $f_2 = -3$, $c_2 = 11$
Step 3: Analysis
Substitute the parameters into the orthogonality condition: \[ 2(1)(-2) + 2(-1)(-3) = c + 11 \] \[ -4 + 6 = c + 11 \] \[ 2 = c + 11 \implies c = -9 \]
Step 4: Conclusion
The value of $c$ is $-9$.
Final Answer: (A)
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