Question

The equation of a circle in general form is ​x2+y2+20x+12y+15=0​ .

What is the equation of the circle in standard form?



(x+10)2+(y+6)2=121


(x+10)2+(y+6)2=225


(x+20)2+(y+12)2=121


(x+20)2+(y+12)2=225

Answer 1

`(x+10)^2+(y+6)^2=121`

Explanation:

The standard form of a circle's equation is

`(x-h)^2 + (y-k)^2 = r^2`

Where (h,k) is the center and r is the radius of the circle.

The equation is given in general form, we need to convert it to the standard form. This can be done by completing squares.

The equation is:

`x^2+y^2+20x+12y+15=0`

Rearrange x's and y's separate:

`x^2+20x+y^2+12y=-15`

We have also subtracted 15. Now to complete squares, recall the formula:

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

The first two terms are equivalent to the first two terms of the above formula. It's clear that the second term is 2ab=20x. Solving for b:

b=10. Thus, to complete the square, we add 100 to the equation.

The third and fourth terms can also be used to complete the squares for y, knowing that 2ab=12y. In this case:

b=6. To complete this square, we add 36. The equation is now:

`(x^2+20x+100)+(y^2+12y+36)=-15+100+36=121`

Operating and simplifying, knowing that 121=11^2:

`\boxed{\mathbf{(x+10)^2+(y+6)^2=121}}`

This corresponds to the first choice