Question:
If the circle $x^2 + y^2 - 4x - 6y + \lambda = 0$ touches the x-axis, then the value of $\lambda$ is:
If the circle $x^2 + y^2 - 4x - 6y + \lambda = 0$ touches the x-axis, then the value of $\lambda$ is:
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Standard relations: touching the x-axis means $g^2 = c$. Touching the y-axis means $f^2 = c$.
Updated On: May 31, 2026
- $4$
- $9$
- $13$
- $16$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
A circle $x^2 + y^2 + 2gx + 2fy + c = 0$ touches the x-axis if and only if $g^2 = c$.
Step 2: Meaning
Comparing the given circle $x^2 + y^2 - 4x - 6y + \lambda = 0$ with the general equation, we get $2g = -4 \implies g = -2$, and $c = \lambda$.
Step 3: Analysis
Using the condition for the circle touching the x-axis: \[ g^2 = c \implies (-2)^2 = \lambda \implies \lambda = 4 \]
Step 4: Conclusion
The value of the parameter $\lambda$ is $4$. Final Answer: (A)
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