Coordinate Geometry of the Circle

Louise Boyle, Oct 8, 2024

Coordinate Geometry of The Circle

Table of Contents

  1. Terminology
    1. LB Circle Terminology
  2. Circle with centre (0,0) and radius r
    1. LB Circle Equation: Center (0,0)
    2. LB Circle Equation: Center (0,0)- Position of a point
  3. Circle with centre (h,k) and radius r
    1. LB The Circle Centre (h,k) Radius r

1.

Terminology

  • 1. LB Circle Terminology

1.1.

LB Circle Terminology

[color=#000000]There are many vocabulary terms we use when talking about a circle. [br]The following app was designed to help you clearly see and interact with each term. [br][br]Explore these apps for a few minutes. [br]As you go along, please answer the questions that follow. [/color]
Note: LARGE POINTS are moveable.
How would you describe or define a [b]CIRCLE[/b] as a set of points that meets specified criteria?
How would you describe the term [b][color=#38761d]RADIUS[/color] [/b][i]without using the words "half" or "diameter" [/i]in your description?
Note: LARGE POINTS are moveable.
What does the term [b][color=#9900ff]CHORD[/color] [/b]mean here in the context of a circle?
How would you describe the term [b][color=#ff7700]DIAMETER[/color] [/b][i]without using the words "two", "double", or "diameter" [/i]in your description?
Note: LARGE POINTS are moveable.
How would you describe/define the term [b][color=#cc0000]SECANT[/color][/b]?
Note: LARGE POINTS are moveable.
What does it mean for a line to be [b][color=#1e84cc]TANGENT [/color][/b]to a circle?
– Louise Boyle, Tim Brzezinski

2.

Circle with centre (0,0) and radius r

  • 1. LB Circle Equation: Center (0,0)
  • 2. LB Circle Equation: Center (0,0)- Position of a point

2.1.

LB Circle Equation: Center (0,0)

For the questions below, be sure to zoom out if you need to!
1.
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius 5. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
2.
What is the equation of a circle with center (0,0) and radius [i]r[/i] = 9?
3.
Suppose another circle has center (0,0). Suppose this circle also passes through the point (12, -5).[br]Write the equation of this circle. [br]
4. FINAL QUESTION:
Suppose [i]P(x,y)[/i] = any point that lies on a circle with center (0,0) and radius [i]r[/i], where [i]r[/i] > 0. [br]Use what you've observed to write an equation that expresses the relationship among [i]x[/i], [i]y[/i], and [i]r[/i].
Quick (Silent) Demo
– Louise Boyle, Tim Brzezinski

2.2.

–

3.

Circle with centre (h,k) and radius r

  • 1. LB The Circle Centre (h,k) Radius r

3.1.

LB The Circle Centre (h,k) Radius r

Move the centre of the circle to (1,1) and make the radius 4.[br]What do you notice about the equation of that circle?
Play around with the applet by changing the centre and radius while paying particular attention to the equation of the corresponding equation of the circle.[br][br]What do you notice?
General equation of a circle
What is the general equation of a circle has centre (h,k) and radius r?
– Louise Boyle

Information