Question

Identify the standard form of a circle equation x^2+2x+y^2-2y=2

Answer 1

The standard from of the expression is:
`(x +1)^2 + (y-1)^2  = (2)^2`

Explanation:

Here the given expression is :

`x^2+2x+y^2-2y=2`

Now, the standard form of a circle is given as :

`(x-h)^2 + (y -k)^2  = r^2`

Here, (h,k) = Coordinates of Center, r = Radius

Also, use the algebraic identity:

`(a \pm b)^2 = a^2 + b^2 \pm 2ab\\`

Now, converting the given expression in the standard form, we get:

`x^2+2x+y^2-2y=2`

Add 2 on both sides of the equation, we get:

`x^2+2x+y^2-2y  +2=2  + 2\\\implies x^2+2x + 1 +  y^2-2y + 1  = 4\\\implies  (x^2+2x + 1 )+  (y^2-2y + 1)  = 4\\\implies (x +1)^2 + (y-1)^2  = (2)^2`

So, here the standard from of the expression is:

`(x +1)^2 + (y-1)^2  = (2)^2`

Center coordinates here are (h,k) = ( -1 ,1) and Radius = 2 units