Question

How do you find the standard form of `x^2 + y^2 + 8x + 2y - 8 = 0` and what kind of a conic is it?

Answer 1

Shape: Circle
`(x+4)^2 + ( y+1)^2 = 9`

Explanation:

Remember: Some of the formula for conic are:
Circle: `(x-h)^2 + (y-k)^2 = r^2`
Ellipse: `(x-h)^2/a^2 + (y-k)^2/b^2 = 1`
Hyperbola:`(x-h)^2/a^2 - (y-k)^2/b^2 = 1`

Step 1: Complete the so to determine the form

`x^2 + 8x + y^2 +2y = 8`

`(x^2 + 8x+color(red)(16)) + (y^2 + 2y+color(red)(1)) = -8+ color(red)(16+ 1`

*Note: to complete the square, `color(red)(c= (b/2)^2`
`(x+4)^2 + ( y+1)^2 = 9`

Center: `(-4, -1)`
Radius: 3