Question

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

part 1 of 2
a. type the equation in center-radius form.
(simplify your answer.)

Explanation:

🆕 New Concept Discovered: Equations of Circles
The standard way to describe a circle using its center and radius.

Step 1: Identify the center and radius from the graph

By looking at the given graph, we can identify key points on the circle:

• The leftmost point is at \((-10, 4)\).
• The rightmost point is at \((0, 4)\).
• The highest point is at \((-5, 9)\).
• The lowest point is at \((-5, -1)\).

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

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

$$ h = \frac{-10 + 0}{2} = -5 $$

$$ k = 4 $$

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

The radius \(r\) is the distance from the center to any point on the circle:

• Distance from the center \((-5, 4)\) to the rightmost point \((0, 4)\):

$$ r = 0 - (-5) = 5 $$

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 set the equation to zero:

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

Combine like terms and subtract \(25\) from both sides:

$$ x^2 + y^2 + 10x - 8y + 25 + 16 - 25 = 0 $$

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

Answer:

a. Center-radius form:

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

b. General form:

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