Circle equation question

[Crosshash]

Homework Statement

A circle has equation x² + y² - 8x - 4y = 9

(i) Show that the centre of this circle is (4,2) and find the radius of the circle.

Homework Equations

Circle equation = (x-a)² + (x-b)² = r²

The Attempt at a Solution

Well, if the centre of the circle is (4,2). Then the equation will be something like:

(x-4)² + (y-2)² = 9

except, if I expand out the two brackets, I get

(x² -8x + 16) + (y² -4y + 4) = 9

x² - 8x + y² -4y + 20 = 9

Here is where i get confused

So (i'm just double checking here since I don't have the answers), is the radius of the circle sqr9 or sqr 11? And is this really showing that (4,2) is the centre?

Thanks

 

[Hootenanny]

Well, if the centre of the circle is (4,2). Then the equation will be something like:

(x-4)² + (y-2)² = 9

except, if I expand out the two brackets, I get

(x² -4x + 16) + (y² -2y + 4) = 9

x² - 4x + y² -2y + 20 = 9

Here is where i get confused

You have expanded the brackets incorrectly.

 

[Crosshash]

You have expanded the brackets incorrectly.

oops, fix'd (i hope :) )

 

[DeanBH]

Homework Statement

A circle has equation x² + y² - 8x - 4y = 9

(i) Show that the centre of this circle is (4,2) and find the radius of the circle.

Homework Equations

Circle equation = (x-a)² + (x-b)² = r²

The Attempt at a Solution

Well, if the centre of the circle is (4,2). Then the equation will be something like:

(x-4)² + (y-2)² = 9

except, if I expand out the two brackets, I get

(x² -8x + 16) + (y² -4y + 4) = 9

x² - 8x + y² -4y + 20 = 9

Here is where i get confused

So (i'm just double checking here since I don't have the answers), is the radius of the circle sqr9 or sqr 11? And is this really showing that (4,2) is the centre?

Thanks

Don't let the post above you put you off. You've expanded right but your fundamentally wrong.

I'll explain this simply.

x² + y² - 8x - 4y = 9

ok you have to make the factorization to reach the -8x and the -4y

which is (x-4)^2 + (y-2)^2 = 9 which is correct.

Though this isn't finished, because when you factorized into those brackets you also added an extra -4^2 and a -2^2 that you didn't need. So to balance this, you have to add these to the right hand side of the equation.

it ends up : (x-4)^2 + (y-2)^2 = 9 + 4^2 + 2^2
= 29

your radius is, for some reason, the square root of 29

 

[Hootenanny]

Don't let the post above you put you off. You've expanded right but your fundamentally wrong.

No he didn't expand the brackets correctly, notice that he edited his post after I posted. See my quoted text above.

 

[DeanBH]

No he didn't, notice that he edited his post after I posted. See my quoted text above.

sorry, I misread your post.

I thought you told him he expanded correctly.

 

[Crosshash]

Thanks for the replies, I managed to reach sqr29 using a different method though.

x² + y² - 8x - 4y = 9

so

x² + y² - 8x - 4y - 9 = 0

and the equation of a circle is

(x - a)² + (x - b)² = r²

expand out

x² + y² - 2ax - 2by + a² + b² - r²

equating coeficients

-2a = -8
a = 4

-2b = -4
b = 2

constant terms

a² + b² - r² = -9

4² + 2² - r² = -9

-r² = -29

r = sqr29

and I think that shows that the centre is (4,2) as well

This way seems logical but much longer