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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.
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 x^{2} and x terms only.
3
Complete the square on these terms.
- Replace the x^{2} 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).
1^{2} = 1 × 1 = 1Write it in the gap after the − sign.
4
Consider the y^{2} and y terms only.
5
Complete the square on these terms.
- Replace the y^{2} 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).
2^{2} = 2 × 2 = 4Write it in the gap after the − sign.
6
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 x^{2} + y^{2} − 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
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