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Circle equations

Mathematics - Grade 11 · Analytical Geometry

Circle Equations

A circle is a set of points in a plane that are equidistant from a fixed point called the centre. The distance from the centre to any point on the circle is called the radius. Understanding the equation of a circle is essential in analytical geometry.

The Standard Equation of a Circle

The standard equation of a circle with centre at point (h, k) and radius r is given by:

(x - h)^2 + (y - k)^2 = r^2

Remember: The coordinates (h, k) represent the centre of the circle, and r is the radius.

Finding the Equation of a Circle

To find the equation of a circle, you need the centre and the radius. Let’s go through an example.

Example 1

Find the equation of a circle with centre at (3, -2) and a radius of 5.

  1. Identify the centre (h, k): (3, -2)
  2. Identify the radius r: 5
  3. Substitute h, k, and r into the standard equation:
(x - 3)^2 + (y + 2)^2 = 5^2

This simplifies to:

(x - 3)^2 + (y + 2)^2 = 25

Thus, the equation of the circle is:

(x - 3)^2 + (y + 2)^2 = 25

Graphing a Circle

To graph a circle, you need to identify the centre and the radius. The centre is plotted first, and then points are plotted at a distance equal to the radius in all directions from the centre.

Example 2

Graph the circle with the equation (x + 1)^2 + (y - 3)^2 = 16.

  1. Identify the centre: (h, k) = (-1, 3)
  2. Identify the radius: r = √16 = 4
  3. Plot the centre at (-1, 3).
  4. From the centre, plot points 4 units away in all directions:
    • Right: (-1 + 4, 3) = (3, 3)
    • Left: (-1 - 4, 3) = (-5, 3)
    • Up: (-1, 3 + 4) = (-1, 7)
    • Down: (-1, 3 - 4) = (-1, -1)

Connect these points in a smooth, round shape to complete the circle.

Converting General Form to Standard Form

The general form of a circle's equation is given by:

Ax^2 + Ay^2 + Bx + Cy + D = 0

To convert this to standard form, follow these steps:

  1. Group the x and y terms:
    2. Ax^2 + Bx + Ay^2 + Cy + D = 0
  2. Factor out A from the x and y terms:
    3. A(x^2 + (B/A)x + y^2 + (C/A)y) + D = 0
  3. Complete the square for both x and y terms.

Example 3

Convert the equation 2x^2 + 2y^2 - 8x + 6y - 10 = 0 to standard form.

  1. Group the terms:
    2. 2(x^2 - 4x) + 2(y^2 + 3y) - 10 = 0
  2. Factor out 2:
    3. 2(x^2 - 4x + y^2 + 3y) = 10
  3. Complete the square:
    • For x^2 - 4x, add and subtract (4/2)^2 = 4:
      • (x^2 - 4x + 4 - 4) = (x - 2)^2 - 4
    • For y^2 + 3y, add and subtract (3/2)^2 = 2.25:
      • (y^2 + 3y + 2.25 - 2.25) = (y + 1.5)^2 - 2.25
  4. Substituting back gives:
    5. 2((x - 2)^2 - 4 + (y + 1.5)^2 - 2.25) = 10
  5. Distributing and simplifying leads to:
    6. (x - 2)^2 + (y + 1.5)^2 = 25

The standard form is:

(x - 2)^2 + (y + 1.5)^2 = 25

Common Mistakes

Watch out: When completing the square, ensure you add and subtract the same value to keep the equation balanced.

Summary

  • The standard equation of a circle is (x - h)^2 + (y - k)^2 = r^2.
  • To graph a circle, plot the centre and then plot points at the radius distance in all directions.
  • To convert from general to standard form, group terms, complete the square, and simplify.

Check your understanding

  1. What is the standard equation of a circle with centre (2, 3) and radius 6?
  2. Convert the equation x^2 + y^2 - 6x + 8y + 9 = 0 to standard form.
  3. Graph the circle with the equation (x - 4)^2 + (y + 1)^2 = 9.
  4. Identify the centre and radius of the circle from the equation (x + 5)^2 + (y - 2)^2 = 36.