1. ## Circle equation, centre and radius

Hi! I am new here and need some help. I am doing a maths course at home through Open Uni and I have got stuck.

The equation

x^2 + 10x + y^2 - 6y + 9 = 0

represents a circle. Find the centre and it's radius.

I have had a look around online at other examples and it says to put it in a certain format with h,k & r. But I just can't follow it to make the equation given here fit in. Can anyone give me some guidance and help working it out please?

2. Hi

Let C be the circle center $\displaystyle A(x_A,y_A)$ and radius R

A point $\displaystyle M(x,y)$ is on the circle if and only if
$\displaystyle AM = R$ which is equivalent to $\displaystyle AM^2 = R^2$

Going through coordinates
$\displaystyle (x-x_A)^2 + (y-y_A)^2 = R^2$

$\displaystyle x^2 + 10x + y^2 - 6y + 9 = 0$
can be written as
$\displaystyle (x+5)^2 - 25 + (y-3)^2 - 9 + 9 = 0$
$\displaystyle (x+5)^2 + (y-3)^2 = 25 = 5^2$

Now I think that you can find the center and the radius

3. Originally Posted by Bunnykins87
Hi! I am new here and need some help. I am doing a maths course at home through Open Uni and I have got stuck.

The equation

x^2 + 10x + y^2 - 6y + 9 = 0

represents a circle. Find the centre and it's radius.

I have had a look around online at other examples and it says to put it in a certain format with h,k & r. But I just can't follow it to make the equation given here fit in. Can anyone give me some guidance and help working it out please?
A circle with the center C(h, k) and the radius r has the equation

$\displaystyle (x-h)^2+(y-k)^2=r^2$

I'll show you the steps to transform your equation into the form of the general equation of a circle:
\displaystyle \begin{aligned}x^2 + 10x + y^2 - 6y + 9 &= 0 \\ x^2 + 10x + y^2 - 6y &=- 9 \\ x^2+10x+25 + y^2 - 6x +9 &=-9+25+9\\ (x+5)^2+(y-3)^2&=5^2\end{aligned}

Thus the given equation describes a circle with C(-5, 3) and r = 5

4. Originally Posted by Bunnykins87
I have had a look around online at other examples and it says to put it in a certain format with h,k & r. But I just can't follow it to make the equation given here fit in. Can anyone give me some guidance and help working it out please?
If you are having trouble rewriting the equation in the appropriate form, you should familiarize yourself with the method of completing the square. That will make these types of problems easy, and it is very useful to know in general.

5. Originally Posted by Bunnykins87
Hi! I am new here and need some help. I am doing a maths course at home through Open Uni and I have got stuck.

The equation

x^2 + 10x + y^2 - 6y + 9 = 0

represents a circle. Find the centre and it's radius.

I have had a look around online at other examples and it says to put it in a certain format with h,k & r. But I just can't follow it to make the equation given here fit in. Can anyone give me some guidance and help working it out please?
Hi, I'm in year 11 doing AS level maths.
Complete the square on it so it becomes
(x+5)^2 + (y-3)^2
expand it to show the numbers at the end
-> (5x5) +25 and (-3x-3) +3
x^2 + 10x + 25 + y^2 - 6y + 9 = 25
[the number after the = is 19 because the right side numbers add up to
34 and 34-9 = 25]
so r^2 = 25
therefore, r = 5
so therefore, the centre (h,k) = (-5, 3) [because x+5 = 0 so x must be 5 and y-3 = 0 so y must be 3)
so h = -5 and k = 3

[I think this is right]