# 9.3 – Circles with Coordinates and Proofs

## Key Terms

• Circle – All points on a plane that are the same distance from a single point.
• Center of the Circle – The point exactly in the middle of a circle.
• Origin – Where x and y coordinates are both zero on a coordinate plane.
• Written as (0,0)
• Radius – The distance from the center of a circle to any point on the circle.
• The size of a circle is defined by its radius.

## Review

Solutions for Lines and Circles
• The solution set for the equation of a line is all the points that lie on the line.
• The solution set for the equation of a circle is all the points that lie on the circle.
• Circle Formula and Coordinates

• Unit Circle

## Notes

Distance Formula: Used to Find the Radius of a Circle
• $D=\sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}}$
• $r=\sqrt{(x-0)^{2}+(y-0)^{2}}$
• $r=\sqrt{x^{2}+y^{2}}$
Standard Equation for Circle Centered at Origin Standard Equation for Any Circle
• $x^{2}+y^{2}=r^{2}$
• r = radius of the circle
• x = x-coordinate of point on the circle
• y = y-coordinate of point on the circle

• Ex. Circle centered at the origin with radius 8
• $x^{2}+y^{2}=8^{2}$
• $x^{2}+y^{2}=64$
• $(x-h)^{2}+(y-v)^{2}=r^{2}$
• r = radius of the circle
• (x, y) = a point on the circle
• (h, v) = center of the circle
• h: horizontal (x-coordinate)
• v: vertical (y-coordinate)

• Ex. Circle centered at (-3,5) with radius 5
• $(x-(-3))^{2}+(y-(5))^{2}=5^{2}$
• $(x+3)^{2}+(y-5)^{2}=25$
Moving the Circle Graphing and Writing the Equation of a Circle
• When you move the circle left or right, the number that changes in its equation is the x-coordinate of the center.
• When you move the circle up or down, the number that changes in its equation is the y-coordinate of the center.
• To write the equation for any circle, you need to know its radius and the coordinates of its center.
• You can use the equation of a circle to graph the circle on an xy-plane and find the coordinates of points on the circle.
• Each xy-plane has 4 quadrants
• Top right: (+,+)
• $(x-h)^{2}+(y-v)^{2}=r^{2}$
• Top left: (-,+)
• $(x+h)^{2}+(y-v)^{2}=r^{2}$
• Bottom left: (-,-)
• $(x+h)^{2}+(y+v)^{2}=r^{2}$
• Bottom right: (+,-)
• $(x-h)^{2}+(y+v)^{2}=r^{2}$
• Circles Centered on the x-axis
• $(x-0)^{2}+(y-v)^{2}=r^{2}$
• $(x-0)^{2}+(y+v)^{2}=r^{2}$
• Circles Centered on the y-axis
• $(x-h)^{2}+(y-0)^{2}=r^{2}$
• $(x+h)^{2}+(y-0)^{2}=r^{2}$

• Ex. The following circles have their centers in the second quadrant.
• $(x+4)^{2}+(y-4)^{2}=7$
• $(x+11)^{2}+(y-8)^{2}=18$
• Ex. The following circles have their centers in the third quadrant.
• $(x+16)^{2}+(y+3)^{2}=17$
• $(x+9)^{2}+(y+12)^{2}=36$

## Examples

 Ex 1. This circle is centered at the origin, and the length of its radius is 6. What is the circle’s equation? Answer: $x^{2}+y^{2}=36$ Ex 2. The equation for the circle below is $x^{2}+y^{2}=81$. What is the length of the circle’s radius? Answer: 9 Ex 3. The circle below is centered at the point (4, 1) and has a radius of length 2. What is its equation? Answer: $(x-4)^{2}+(y-1)^{2}=4$ Ex 4. The circle below is centered at the point (4, -3) and has a radius of length 5. What is its equation? Answer: $(x-4)^{2}+(y+3)^{2}=5^{2}$ Ex 5. What is the equation of the circle with center (-3.2, -2.1) and radius 4.3? Answer: $(x+3.2)^{2}+(y+2.1)^{2}=4.3^{2}$ Ex 6. The following circles lie completely within the third quadrant, not just the center. Graph them to find out for sure. $(x+12)^{2}+(y+9)^{2}=9$ $(x+7)^{2}+(y+7)^{2}=4$