Question

If the circle $x^2 + y^2 - 4x - 6y + \lambda = 0$ touches the x-axis, then the value of $\lambda$ is:

• A. $4$
• B. $9$
• C. $13$
• D. $16$

Hint:

Standard relations: touching the x-axis means $g^2 = c$. Touching the y-axis means $f^2 = c$.

Answer 1

The Correct Option is A

Step 1: Touching the x-axis condition.
A circle touches the x-axis when the distance from its centre to that axis equals its radius. For $x^2+y^2+2gx+2fy+c=0$ this becomes $g^2 = c$.

Step 2: Match coefficients.
From $x^2 + y^2 - 4x - 6y + \lambda = 0$: $2g = -4$ so $g = -2$, and $c = \lambda$.

Step 3: Apply the condition.
\[ g^2 = c \implies (-2)^2 = \lambda \]

Step 4: Compute.
\[ 4 = \lambda \]

Step 5: Quick check.
Centre is $(2,3)$, radius is $\sqrt{4 + 9 - 4} = 3$, which equals the $y$-coordinate of the centre, confirming it touches the x-axis.

Step 6: Conclusion.
\[ \boxed{ \lambda = 4 } \]