x^2 + y^2 + 8x + 6y= 0

We need to re-write the equation in the following format:

( x- a)^2 + ( y-b)^2 = r^2   such that:

(a,b) is the center of the circle and r is the radius:

Let us calculate:

We will re-arrange terms:

x^2 + 8x + y^2 + 6y = 0

We will complete the squares:

==> ( x+4)^2 - 16 + ( y+3)^2 - 9 = 0

==> ( x+4)^2 + ( y+3)^2 - 25 = 0

We will move 25 to the right side:

==> ( x+ 4)^2 + ( y+3)^2 = 25

==> ( x+ 4)^2 + ( y+3)^2 = 5^2

Now the equation is in the standard form:

Then the center of the circle is ( -4, -3) and the radius = 5

We have the equation of the circle given as x^2 + y^2 + 8x + 6y = 0.

Now, the equation of a circle with center (h, k) and radius r is (x − h)^2 + (y − k)^2 = r^2

So we convert x^2 + y^2 + 8x + 6y = 0 to the standard form.

x^2 + y^2 + 8x + 6y = 0

=> x^2 + 8x + y^2 + 6y = 0

=> x^2 + 8x + 16 + y^2 + 6y + 9 = 16 + 9

=> (x + 4)^2 + (y +3)^2 = 25 = 5^2

So the center is ( -4, -3) and the radius is 5.

The centre and radius of the circle x^2 + y^2 + 8x + 6y = 0 is ( -4, -3) and 5 respectively.