# How to Convert an Equation of a Circle from General to Standard Form

## Converting an Equation of a Circle from General to Standard Form

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. # 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

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 centered at (1, 2) with a radius of 9. 