Question

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

Explanation:

🆕 New Concept Discovered: Equations of Circles
Finding center, radius, and converting forms.

Step 1: Identify the Center and Radius

To find the equation of a circle, we need its center \((h, k)\) and its radius \(r\).

Looking at the graph, the four outermost points (vertices) of the circle are given:

• Top: \((-5, 9)\)
• Bottom: \((-5, -1)\)
• Left: \((-10, 4)\)
• Right: \((0, 4)\)

The center \((h, k)\) is the midpoint of the horizontal or vertical diameter:

• The horizontal diameter connects \((-10, 4)\) and \((0, 4)\). The midpoint is:

$$ h = \frac{-10 + 0}{2} = -5, \quad k = 4 $$

• Thus, the center is \((h, k) = (-5, 4)\).

The radius \(r\) is the distance from the center to any of these outer points:

• Distance from center \((-5, 4)\) to the top point \((-5, 9)\):

$$ r = 9 - 4 = 5 $$

Step 2: Write the Center-Radius Form

The standard center-radius form of a circle's equation is:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

Substitute \(h = -5\), \(k = 4\), and \(r = 5\):

$$ (x - (-5))^2 + (y - 4)^2 = 5^2 $$

$$ (x + 5)^2 + (y - 4)^2 = 25 $$

Step 3: Convert to General Form

To find the general form, expand the squared binomials and move all terms to one side so the equation equals \(0\):

$$ (x^2 + 10x + 25) + (y^2 - 8y + 16) = 25 $$

Combine like terms:

$$ x^2 + y^2 + 10x - 8y + 41 = 25 $$

Subtract \(25\) from both sides:

$$ x^2 + y^2 + 10x - 8y + 16 = 0 $$

Answer:

(a) Center-radius form:

$$ (x + 5)^2 + (y - 4)^2 = 25 $$

(b) General form:

$$ x^2 + y^2 + 10x - 8y + 16 = 0 $$