Question

1. write the equation of the circle to the right in standard form.

$\left(\quad\
ight)^2 + \left(\quad\
ight)^2 = \quad$

1. write the equation of the circle to the right in standard form.

$\left(\quad\
ight)^2 + \left(\quad\
ight)^2 = \quad$

Explanation:

Problem 9

Step1: Recall circle standard form

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step2: Identify center of circle 9

From the graph, the center of the green circle is at \((0, 5)\), so \(h = 0\), \(k = 5\).

Step3: Determine radius of circle 9

Looking at the graph, the circle extends from \(x = -10\) to \(x = 10\) (or similar, but visually the radius seems 10? Wait, no, center at (0,5), and the circle goes to x=10? Wait, no, let's check. Wait, the grid: from center (0,5), the distance to the rightmost point (10,5)? Wait, no, the green circle: looking at the x-axis, the circle goes from -10 to 10? Wait, no, the center is (0,5), and the radius: let's see, the circle's rightmost point is at x=10, y=5? Wait, no, the graph: the green circle, center at (0,5), and the radius: from center (0,5) to (10,5) is 10 units? Wait, no, maybe I misread. Wait, the center is (0,5), and the circle's radius: let's check the y-axis. The circle goes up to y=15? No, the grid has 10, 5, -5, -10. Wait, center at (0,5), and the circle's top is at y=15? No, the green circle: center (0,5), and the radius: from (0,5) to (10,5) is 10? Wait, no, the x-coordinate of the center is 0, and the rightmost point is at x=10, so radius \(r = 10\) (since distance from 0 to 10 is 10). Wait, no, maybe the radius is 10? Wait, no, let's recalculate. Wait, the standard form: \((x - 0)^2 + (y - 5)^2 = r^2\). Now, find \(r\). The circle's rightmost point is at (10,5), so distance from (0,5) to (10,5) is \(10 - 0 = 10\), so \(r = 10\). Wait, but then \(r^2 = 100\). Wait, but maybe I made a mistake. Wait, no, let's check again. Wait, the center is (0,5), and the circle: from x=-10 to x=10, so diameter is 20, radius 10. So \(h=0\), \(k=5\), \(r=10\). So the equation is \((x - 0)^2 + (y - 5)^2 = 10^2\), so \((x)^2 + (y - 5)^2 = 100\). Wait, but maybe I messed up the center. Wait, the green circle: the center is at (0,5), yes. So \(h=0\), \(k=5\), \(r=10\).

Step1: Recall circle standard form

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step2: Identify center of circle 10

From the graph, the center of the purple circle is at \((5, -5)\), so \(h = 5\), \(k = -5\).

Step3: Determine radius of circle 10

Looking at the graph, the circle's radius: from center (5, -5) to (10, -5) is 5 units? Wait, the purple circle: center at (5, -5), and the radius: let's see, the circle goes from x=0 to x=10? Wait, center at (5, -5), so distance to x=0 is 5, to x=10 is 5. So radius \(r = 5\), so \(r^2 = 25\).

Answer:

\((x - 0)^2 + (y - 5)^2 = 100\) (or \((x)^2 + (y - 5)^2 = 100\))

Problem 10