Key Concepts and Formulas

• Equation of a Circle: The general equation of a circle is given by $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $r = \sqrt{g^2 + f^2 - c}$.
• Area of an Equilateral Triangle Inscribed in a Circle: If an equilateral triangle is inscribed in a circle of radius $r$, its area is given by $\frac{3\sqrt{3}}{4}r^2$.
• Sine of 120 degrees: $\sin(120^\circ) = \frac{\sqrt{3}}{2}$.

Step-by-Step Solution

Step 1: Determine the radius of the circle using the given area of the inscribed equilateral triangle.

We are given that the area of the equilateral triangle is $27\sqrt{3}$. We will use the formula for the area of an equilateral triangle inscribed in a circle to find the radius. $\text{Area} = \frac{3\sqrt{3}}{4}r^2$ We substitute the given area: $27\sqrt{3} = \frac{3\sqrt{3}}{4}r^2$ Why? We know the area and want to find the radius ($r$), so we plug in and solve. Now, we solve for $r^2$: $r^2 = \frac{27\sqrt{3} \cdot 4}{3\sqrt{3}}$ $r^2 = \frac{27 \cdot 4}{3}$ $r^2 = 9 \cdot 4$ $r^2 = 36$ Thus, $r = 6$.

Step 2: Express the radius of the given circle in terms of $c$.

The equation of the circle is given as $x^2 + y^2 + 10x + 12y + c = 0$. We need to express the radius in terms of $c$ using the general equation of a circle. Why? The problem gives the circle equation and asks for $c$, so we need to relate the radius to $c$ using the given equation. Comparing this equation with the general form $x^2 + y^2 + 2gx + 2fy + k = 0$, we have: $2g = 10 \implies g = 5$ $2f = 12 \implies f = 6$ $k = c$ The radius is given by $r = \sqrt{g^2 + f^2 - c}$. Therefore, $r^2 = g^2 + f^2 - c$. Substituting the values of $g$ and $f$, we get: $r^2 = 5^2 + 6^2 - c$ $r^2 = 25 + 36 - c$ $r^2 = 61 - c$

Step 3: Equate the two expressions for $r^2$ and solve for $c$.

From Step 1, we have $r^2 = 36$. From Step 2, we have $r^2 = 61 - c$. We equate these two expressions to solve for $c$. Why? Since both expressions represent the square of the radius of the same circle, they must be equal. $36 = 61 - c$ Solving for $c$: $c = 61 - 36$ $c = 25$

Tips and Common Mistakes to Avoid:

• Be careful with the signs when determining the center of the circle from the general equation. The center is $(-g, -f)$, not $(g, f)$.
• Remember the formula for the area of an equilateral triangle inscribed in a circle. It is crucial for relating the given area to the radius.
• Avoid algebraic errors when solving for $r^2$ and $c$. Double-check your calculations.

Summary

We first found the radius of the circle using the area of the inscribed equilateral triangle. Then, we expressed the radius in terms of $c$ using the given equation of the circle. Finally, we equated the two expressions for the radius squared to solve for $c$.

The final answer is \boxed{25}, which corresponds to option (B).