CIRCLES: WRITING EQUATION OF A CIRCLE

[Pages:15]CIRCLES: WRITING EQUATION OF A CIRCLE

Unit Overview

In this unit, students will identify the center and radius of a circle and write the standard form of the equation of a circle.

Key Vocabulary

Circle

Center Radius Diameter

Standard Form of a Circle

Set of all points at a specific distance from a given point in two dimensions. Point that defines the location of the circle. Distance from the center of the circle to the edge of the circle. Straight line segment that passes through the center of the circle and whose endpoints lie on the circle (x ? h)2 + (y ? k)2 = r2 Standard equation for the circle centered at (h, k) with radius r.

Center of a Circle

Radius of a Circle

Diameter of a Circle

Equation of a Circle

A circle is a set of points in the xy plane that are a fixed distance r from a fixed point (h, k), where (h, k) is the center of the circle and r is the radius. Another way to state this is that a circle is the locus of points in a plane equidistant from a given point. This definition can help you write an equation of a circle.

Look at the image below. You will notice that (h, k) is the center of the circle and (x, y) is a point on the circle. The distance between (h, k) and (x, y) is the length of the radius.

The standard form of a circle tells you two things you need: it tells you where the center is, and it tell you what the radius is.

Before moving on in this unit complete the following assignments.

1.) Review the Key Vocabulary 2.) Review the Equations of a Circle Notes 3.) Click on the word video to get an overview of equations of a circle.

Identify the Center and Radius of Each Circle

Example A ? Answer the following questions for the equation: (x ? 5)2 + (y + 2)2 = 81.

What is the center? It looks like the center would be the point (-5, 2), but you always do the opposite of what it looks like in the equation. The center of the equation is (5, -2).

What is the radius? It looks like the radius would be 81, but the equation is r2, so you have to create the equation r2=81. You take the square root from 81, and the radius is 9.

Example B ? Answer the following questions for the equation: (x + 1)2 + (y + 8)2 = 25

What is the center? Remember to take the opposite of what it looks like in the equation. The center is (-1, -8).

What is the radius? Remember you need to take the square root of the last number in the equation. The square root of 25 is 5. The radius is 5.

Let's Practice ? Identify the Center and Radius

1.) (x ? 5)2 + (y + 3)2 = 9 Center (___, ___) Radius = ___ Center: (5, -3) Radius = 3

2.) (x + 5)2 + (y ? 7)2 = 100 Center (___, ___) Radius = ___ Center: (-5, 7) Radius = 10

3.) (x)2 + (y + 2)2 = 16 Center (___, ___) Radius = ___ Center: (0, -2) Radius = 4

4.) (x ? 1/2)2 + (y + 1/4)2 = 36 Center (___, ___) Radius = ___ Center: (1/2, -1/4) Radius = 6

5.) (x ? 1)2 + (y)2 = 10 Center (___, ___) Radius = ___ Center: (1, 0) Radius = 10 or 3.16

Write an Equation for a Circle

Example C ? Write an equation for a circle: Center = (4, -6) Radius = 5 Step 1: Equation for a circle is (x ? h)2 + (y ? k)2 = r2 Step 2: Plug in the center points and the radius in the equation

(x ? 4)2 + (y ? - 6)2 = 52 Step 3: Simplify and rewrite the equation. Look at the y (two negatives equal a positive) and radius (take the square root of 5) Step 4: Equation is (x ? 4)2 + (y + 6)2 = 25

Example D ? Write an equation for a circle: Center = (3, 1) Diameter = 10 Step 1: Equation for a circle is (x ? h)2 + (y ? k)2 = r2 Look carefully at the equation ? problem gives you the diameter not the radius Step 2: Plug in the center points and the radius in the equation (x ? 3)2 + (y ? 1)2 = 52 (Radius is half of the diameter ? 10/2 = 5) Step 3: Simplify and rewrite the equation. Step 4: Equation is (x ? 3)2 + (y ? 1)2 = 25

Let's Practice ? Write an Equation of a Circle

6.) Write an equation for a circle: Center = (5, -3) Radius = 7 (x ? 5)2 + (y + 3)2 = 49

7.) Write an equation for a circle: Center = (-2, -4) Radius = 6 (x + 2)2 + (y ? 4)2 = 36

8.) Write an equation for a circle: Center = (13, -13) Diameter = 4 (x ? 13)2 + (y + 13)2 = 4

9.) Write an equation for a circle: Center = (-6, --15) Radius = 5 (x + 6)2 + (y + 15)2 = 5

10.) Write an equation for a circle: Center = (12, 0) Diameter = 8 (x ? 12)2 + (y)2 = 16

Identify the Center and Radius Using a Graph

In order to identify the center and radius using a graph, you need to remember the pattern for a circle: (x ? h)2 + (y ? k)2 = r2 in which (h, k) is the center of the circle and r is the radius. Example E ? Identify the center and radius using the graph below.

Find the center of Circle O. Step 1: (h, k) is the center of the circle. h is x-axis and k is y-axis. Step 2: Count the number of units to either right or left of x-axis to find h. h = 1 (move to the right) Step 3: Count the number of units up or down of y-axis to find k. k = 3 (move up) Step 4: The center of the circle is (1, 3)

Find the radius of Circle O. Step 1: Identify the center of the circle. (1, 3) Step 2: Count the number of units to the right, left, up or down to find the radius from center. Step 3: The number is always going to be positive. r = 3 Step 4: The radius of the circle is 3.

Let's Practice ? Identity the Center and Radius Using a Graph

11.) Center ( ___, ___) Radius ___

Center: (1, -3) Radius: 2 12.) Center ( ___, ___) Radius ___

Center: (2, -1) Radius: 4

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