Questions: Use the graph below to determine the equation of the circle in (a) center-radius form and (b) general form.

Use the graph below to determine the equation of the circle in (a) center-radius form and (b) general form.
Transcript text: Use the graph below to determine the equation of the circle in (a) center-radius form and (b) general form.
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Solution

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Solution Steps

Step 1: Identify the Center and Radius of the Circle

From the graph, the center of the circle is at the point (2, 4). The radius can be determined by measuring the distance from the center to any point on the circle. For example, the point (6, 4) is on the circle, and the distance from (2, 4) to (6, 4) is 4 units. Therefore, the radius \( r \) is 4.

Step 2: Write the Equation in Center-Radius Form

The center-radius form of the equation of a circle is: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center and \(r\) is the radius. Substituting the center (2, 4) and radius 4, we get: \[ (x - 2)^2 + (y - 4)^2 = 4^2 \] \[ (x - 2)^2 + (y - 4)^2 = 16 \]

Step 3: Convert to General Form

To convert the center-radius form to the general form, expand and simplify the equation: \[ (x - 2)^2 + (y - 4)^2 = 16 \] \[ (x^2 - 4x + 4) + (y^2 - 8y + 16) = 16 \] Combine like terms: \[ x^2 + y^2 - 4x - 8y + 4 + 16 = 16 \] \[ x^2 + y^2 - 4x - 8y + 20 = 16 \] Subtract 16 from both sides: \[ x^2 + y^2 - 4x - 8y + 4 = 0 \]

Final Answer

  • Center-Radius Form: \((x - 2)^2 + (y - 4)^2 = 16\)
  • General Form: \(x^2 + y^2 - 4x - 8y + 4 = 0\)
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