# Converting an Equation of a Circle from General to Standard Form(KS3, Year 8)

homesitemapgraphs and coordinatesconverting 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.

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

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.

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