The Lesson

We can convert an equation of a circle from general form to standard form.

Why Convert from General Form to Standard Form?

The same circle can be written in general form and standard form. The two forms of equations of a circle tell us different things about the circle.
  • In general form, A, B and C do not tell us anything about the circle.

  • In standard form, • a and b tells us that (a, b) are the Cartesian coordinates of the center of the circle. • r tells us the radius of the circle.
If we are given a circle in general form, we can convert it to standard form to understand more about the circle.

A Real Example of How to Convert an Equation of a Circle from General to Standard Form

Question

Convert the equation of a circle in general form shown below into standard form. Find the center and radius of the circle.

Step-by-Step:

1

Group the x's and y's together.

2

Consider the x2 and x terms only.

3

Complete the square on these terms.
  • Replace the x2 and x terms with a squared bracket. Leave a gap inside the brackets for two terms. Leave a gap after the brackets for a number to be subtracted.

  • Write an x in the brackets.

  • Look at the original equation. Find the sign in front of the x term. In our example, it is −. Write this sign after the x in the brackets.

  • Look at the original equation. Find the number in front of the x term (called the coefficient of x). In our example, it is 2.

    Divide the coefficient of x by 2.
    2 ÷ 2 = 1
  • Write the halved coefficient (1) in the gap in the brackets.

  • Square the halved coefficient (1).
    12 = 1 × 1 = 1
    Write it in the gap after the − sign.

We have completed the square on these terms.

4

Consider the y2 and y terms only.

5

Complete the square on these terms.
  • Replace the y2 and y terms with a squared bracket. Leave a gap inside the brackets for two terms. Leave a gap after the brackets for a number to be subtracted.

  • Write a y in the brackets.

  • Look at the original equation. Find the sign in front of the y term. In our example, it is −. Write this sign after the y in the brackets.

  • Look at the original equation. Find the number in front of the y term (called the coefficient of y). In our example, it is 4.

    Divide the coefficient of y by 2.
    4 ÷ 2 = 2
  • Write the halved coefficient (2) in the gap in the brackets.

  • Square the halved coefficient (2).
    22 = 2 × 2 = 4
    Write it in the gap after the − sign.

We have completed the square on these terms.

6

Add the numbers outside of the brackets together.

− 1 + − 4 + − 4 = − 1 − 4 − 4 = − 9

7

Rearrange the equation so the number is on the right hand side of the equals sign (=).
(x − 1)2 + (y − 2)2 − 9 = 0
(x − 1)2 + (y − 2)2 − 9 + 9= 0 + 9 Add 9 to boths sides
(x − 1)2 + (y − 2)2 = 9

Answer:

We have converted x2 + y2 − 2x − 4y − 4 = 0 (in general form) to (x − 1)2 + (y − 2)2 = 9 (in standard form). This is a circle centred at (1, 2) with a radius of 9.


Read more about how to find the centre and radius from the equation of a centre

Lesson Slides

The slider below gives a real example of how to convert an equation of a circle from general form to standard form.