Question

What is the standard form of 5x^2+5y^2-10x+20y-30=0

Answer 1

`5x^2+5y^2-10x+20y-30=0`

We need to get the equation in square form

We apply completing the square method

`5x^2-10x+5y^2+20y-30=0`

Divide all the terms by 5

`x^2-2x+y^2+4y-6=0`

Group x terms and y terms separately

`(x^2-2x)+(y^2+4y)-6=0`

By completing the square method, we take half of coefficient of x and square it

`(-2)/(2) = -1`

square it
`(-1)^2= 1`

Add 1 on both sides

`(x^2-2x+1)+(y^2+4y)-6=+1`

`(4)/(2) = 2`

square it
`(2)^2= 4`

Add 4 on both sides

`(x^2-2x+1)+(y^2+4y+4)-6=0+1+4`

Add 6 on both sides

`(x^2-2x+1)+(y^2+4y+4)=6+1+4`

Now factor both parenthesis

`(x-1)^2+(y+2)^2=11`

The equation of the circle is
`(x-1)^2+(y+2)^2=11`