1. ## Conic section

I know this is a conic section and that it is a circle, I can get that much but I have forgotten how to simplify it down to a variation of the standard formula, $x^2+y^2=r^2$. I would love for someone to refresh my memory, Thank you.
$
4x^2+y^2-8x+4y+4=0$

2. Originally Posted by OnMyWayToBeAMathProffesor
I know this is a conic section and that it is a circle, I can get that much but I have forgotten how to simplify it down to a variation of the standard formula, $x^2+y^2=r^2$. I would love for someone to refresh my memory, Thank you.
$
4x^2+y^2-8x+4y+4=0$
$4x^2+y^2-8x+4y+4=0$

Group the x and y terms together:

$(4x^2-8x)+(y^2+4y+4)=0\implies 4(x^2-2x)+(y^2+4y+4)=0$

Complete the square for the x term. The equation then becomes:

$4(x^2-2x+1)+(y^2+4y+4)=4$

Now factor:

$4(x-1)^2+(y+2)^2=4$

The equation now becomes $\color{red}\boxed{(x-1)^2+\frac{(y+2)^2}{4}=1}$

This is an ellipse with the major axis along the y-axis.

I hope this makes sense!

--Chris

3. THANK YOU very much, I knew it was something simple like that. It has been so long since i did a problem like that.