Question

If the area of the circle $4x^2 + 4y^2 + 8x - 16y + \lambda = 0$ is $9\pi$ sq. units, then the value of $\lambda$ is:

• A. $4$
• B. $-4$
• C. $16$
• D. $-16$
• E. $-8$

Hint:

Always ensure the coefficients of $x^2$ and $y^2$ are 1 before identifying $g$, $f$, and $c$. Forgetting to divide by 4 in the first step is the most frequent cause of error in this problem type.

Answer 1

The Correct Option is D

Concept: To find the radius and properties of a circle, the equation should first be simplified to the standard form $x^2 + y^2 + 2gx + 2fy + c = 0$. The radius is $r = \sqrt{g^2 + f^2 - c}$.
• Area of a circle $= \pi r^2$.

Step 1: Normalize the circle equation.
The given equation is $4x^2 + 4y^2 + 8x - 16y + \lambda = 0$. Divide by $4$: \[ x^2 + y^2 + 2x - 4y + \frac{\lambda}{4} = 0 \] Comparing with the standard form: $2g = 2 \Rightarrow g = 1$, $2f = -4 \Rightarrow f = -2$, and $c = \frac{\lambda}{4}$.

Step 2: Use the area to find $r^2$.
Area $= 9\pi$. Since Area $= \pi r^2$: \[ \pi r^2 = 9\pi \quad \Rightarrow \quad r^2 = 9 \]

Step 3: Solve for $\lambda$ using the radius formula.
\[ r^2 = g^2 + f^2 - c \] Substitute the values: \[ 9 = (1)^2 + (-2)^2 - \frac{\lambda}{4} \] \[ 9 = 1 + 4 - \frac{\lambda}{4} \quad \Rightarrow \quad 9 = 5 - \frac{\lambda}{4} \] \[ \frac{\lambda}{4} = 5 - 9 \quad \Rightarrow \quad \frac{\lambda}{4} = -4 \] \[ \lambda = -16 \]