Question

The three vertices of a triangle are \( (0,0) \), \( (3,1) \) and \( (1,3) \). If this triangle is inscribed in a circle, then the equation of the circle is

• A. \( 2x^2 + 2y^2 - 2x - 6y = 0 \)
• B. \( x^2 + y^2 - 3x - y = 0 \)
• C. \( x^2 + y^2 - x - 3y = 0 \)
• D. \( 2x^2 + 2y^2 - 6x - 2y = 0 \)
• E. \( 2x^2 + 2y^2 - 5x - 5y = 0 \)

Hint:

For a circle passing through the origin, the constant term \(c\) in the general equation \(x^2+y^2+2gx+2fy+c=0\) is always zero. This instantly simplifies your system of equations, saving valuable time.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
A circle circumscribing a triangle passes through all three of its vertices.
We need to find the general equation of a circle that satisfies the coordinates of the given three points.

Step 2: Key Formula or Approach:
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \] Substitute each vertex \( (x, y) \) into the equation to form a system of linear equations in terms of \( g \), \( f \), and \( c \).

Step 3: Detailed Explanation:
Let the equation of the required circle be:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \quad \dots \text{(Equation 1)} \] The circle passes through the origin \( (0,0) \). Substitute \( x=0, y=0 \) into Equation 1:
\[ 0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \implies c = 0 \] Since \( c = 0 \), the simplified general equation is:
\[ x^2 + y^2 + 2gx + 2fy = 0 \] The circle passes through the point \( (3,1) \). Substitute \( x=3, y=1 \):
\[ 3^2 + 1^2 + 2g(3) + 2f(1) = 0 \] \[ 9 + 1 + 6g + 2f = 0 \] \[ 6g + 2f = -10 \] Dividing the entire equation by 2 gives:
\[ 3g + f = -5 \quad \dots \text{(Equation 2)} \] The circle also passes through the point \( (1,3) \). Substitute \( x=1, y=3 \):
\[ 1^2 + 3^2 + 2g(1) + 2f(3) = 0 \] \[ 1 + 9 + 2g + 6f = 0 \] \[ 2g + 6f = -10 \] Dividing the entire equation by 2 gives:
\[ g + 3f = -5 \quad \dots \text{(Equation 3)} \] Now, we solve Equation 2 and Equation 3 simultaneously.
From Equation 3, isolate \( g \):
\[ g = -5 - 3f \] Substitute this expression for \( g \) into Equation 2:
\[ 3(-5 - 3f) + f = -5 \] \[ -15 - 9f + f = -5 \] \[ -8f = 10 \] \[ f = -\frac{10}{8} = -\frac{5}{4} \] Substitute \( f = -\frac{5}{4} \) back into the isolated equation for \( g \):
\[ g = -5 - 3\left(-\frac{5}{4}\right) \] \[ g = -5 + \frac{15}{4} \] \[ g = \frac{-20 + 15}{4} = -\frac{5}{4} \] Now, substitute the values \( g = -5/4 \), \( f = -5/4 \), and \( c = 0 \) into the general equation:
\[ x^2 + y^2 + 2\left(-\frac{5}{4}\right)x + 2\left(-\frac{5}{4}\right)y = 0 \] \[ x^2 + y^2 - \frac{5}{2}x - \frac{5}{2}y = 0 \] Multiply the entire equation by 2 to clear the fractions:
\[ 2x^2 + 2y^2 - 5x - 5y = 0 \]

Step 4: Final Answer:
The correct equation is found in option (E).