Question

The area of an equilateral triangle inscribed in circle \(x^{2}+y^{2}+2 g x+2 f y+c=0\), is (1) \(\frac{3 \sqrt{3}}{2}\left(g^{2}+f^{2}-c\right)\) (2) \(\frac{3 \sqrt{3}}{4}\left(g^{2}+f^{2}-c\right)\) (3) \(\frac{3 \sqrt{3}}{4}\left(g^{2}+f^{2}+c\right)\) (4) \(\frac{3 \sqrt{3}}{2}\left(g^{2}+f^{2}+c\right)\)

Short answer

Option (2) \( \frac{3\sqrt{3}}{4}(g^2 + f^2 - c) \).

Step-by-step solution

- Convert circle equation to standard form

Rewrite the given circle equation in the standard form. The equation of a circle in general form is given by: \( x^2 + y^2 + 2gx + 2fy + c = 0 \) To convert it to standard form: \( (x + g)^2 + (y + f)^2 = g^2 + f^2 - c \)

- Identify radius

In the standard form of the circle equation, \( (x + g)^2 + (y + f)^2 = r^2 \). Thus, the radius \(r\) of the circle can be identified as: \( r = \sqrt{g^2 + f^2 - c} \)

- Use formula for area of an equilateral triangle

An equilateral triangle inscribed in a circle has a side length equal to \( r\sqrt{3} \), where \( r \) is the radius of the circle. The area \(A\) of an equilateral triangle with side length \(s\) is given by \( A = \frac{\sqrt{3}}{4}s^2 \). For our triangle: \( s = r\sqrt{3} \) Substitute \( s \) into the area formula: \( A = \frac{\sqrt{3}}{4} (r \sqrt{3})^2 = \frac{\sqrt{3}}{4} (3r^2) = \frac{3\sqrt{3}}{4} r^2 \)

- Substitute for the radius

Substitute the value of \( r \) into the area formula: \( r^2 = g^2 + f^2 - c \) \( A = \frac{3\sqrt{3}}{4} (g^2 + f^2 - c) \)

- Verify answer choice

Compare the expression \( \frac{3\sqrt{3}}{4}(g^2 + f^2 - c) \) with the given options. The correct option is: Option (2) \( \frac{3\sqrt{3}}{4}(g^2 + f^2 - c) \)

Key concepts

Circle Equation

Understanding the equation of a circle is fundamental. A circle's equation in its general form is given by: \( x^2 + y^2 + 2gx + 2fy + c = 0 \) To make it easier to work with, we convert this to its standard form. The standard form of a circle equation looks like: \( (x + g)^2 + (y + f)^2 = r^2 \)Here,

• \(g\) and \(f\) are the coordinates of the center of the circle, but with opposite signs, so the center is \((-g, -f)\).
• \(r\) represents the radius of the circle.

To convert the general form to standard form, we complete the square for both \(x\) and \(y\). This results in the equation: \( (x + g)^2 + (y + f)^2 = g^2 + f^2 - c \). This makes it easier to identify the circle’s center and its radius, which leads us to the next concept.

Radius Calculation

Once we have the circle equation in the standard form, identifying the radius becomes straightforward. The standard form resulting from the general equation is: \( (x + g)^2 + (y + f)^2 = g^2 + f^2 - c \) From this, it is clear that the radius squared \(r^2\) is equal to \(g^2 + f^2 - c\). Thus, we find the radius \(r\) by taking the square root of the right-hand side: \( r = \sqrt{g^2 + f^2 - c}\). To summarize:

• Identify \(g^2 + f^2 - c\) from the standard form of the equation.
• Take its square root to find the radius \(r\).

Understanding the radius of the circle is crucial for calculating the dimensions and properties of shapes inscribed within the circle, such as the equilateral triangle discussed next.

Equilateral Triangle Area

An equilateral triangle, by definition, has all three sides of equal length and all internal angles of \(60\) degrees. When an equilateral triangle is inscribed within a circle, its side length can be expressed in terms of the circle’s radius. For an inscribed equilateral triangle, the relationship between the side length \(s\) of the triangle and the circle’s radius \(r\) is: \( s = r \sqrt{3} \). To find the area \(A\) of an equilateral triangle, we use the formula: \( A = \frac{\sqrt{3}}{4} s^2 \). Since the side length \(s\) is \( r \sqrt{3} \), we substitute \(s\) into the area formula: \( A = \frac{\sqrt{3}}{4} (r \sqrt{3})^2 = \frac{\sqrt{3}}{4} (3r^2) = \frac{3\sqrt{3}}{4} r^2 \). Next, we substitute the value of \(r^2\) from the radius equation, which is \( g^2 + f^2 - c \), into the area formula: \( A = \frac{3\sqrt{3}}{4} (g^2 + f^2 - c) \). Thus, the area of the inscribed equilateral triangle depends on the radius of the circle, which ultimately relies on the parameters \(g, f,\) and \(c\) from the general form of the circle's equation. This step-by-step approach ensures the accurate calculation of the triangle's area.