Question

Please help I could not understand how to get the answer.

transform each equation into center - radius form.
1. x2+y2+8x-14y+1=0
answer.
x2-8x=(x-4)2=16
y2=14y=(y+7)2=49
(x-4)2-16+(y7)2-49+40=0

2. y2+x2-2x-12y-12=0
answer

3. x2+y2+4x-5=0


4. y2+x2-6y+5=0


5. x2+y2-256=0

Answer 1

1. x^2+y^2+8x-14y+1=0
we want (x-a)^2 + (y-b)^2 = R^2
write x terms on left
x^2 + 8 x = -y^2 + 14 y - 1
we need to add (half of coef of x) squared to both sides
4*4 = 16
x^2 + 8 x + 16 = -y^2 + 14 y + 15
(x+4)^2 + y^2 - 14 y = 15
now add (half of 14)^2 or 49 squared to each side
(x+4)^2 + y^2 - 14 y + 49 = 15+49 = 64
(x+4)^2 + (y-7)^2 = 15+49 = 64
center at (-4,7) and R = 8
======================
check
when x = -4, y better be +15 or -1
(y-7)^2 = 64
y-7 = +8 or -8
y = 15 or -1 sure enough

Answer 2

now check with that very first line, see if x = -4, y = -1 works
1. x^2+y^2+8x-14y+1=0
16 + 1 - 32 + 14 + 1 = ?? 0 ??
17 - 32 + 15 ?
-15 + 15 = 0 whew !

Answer 3

2. y^2 + x^2 - 2x -12y - 12 = 0
x^2 -2 x = -y^2 + 12 y + 12
add (half of -2)(half of -2) both sides
-1*-1=1
x^2 - 2 x + 1 = -y^2 + 12 y + 13
(x-1)^2 + y^2 - 12 y = 13
add 6^2 or 36 both sides
(x-1)^2 + y^2 - 12 y + 36 = 49
(x-1)^2 + (y-6)^2 = 7^2
center at (1,6) and radius = 7

Answer 4

If you want to make one of these up
say center at (3,4) and radius = 2
that would be
(x-3)^2 + (y-4)^2 = 4
x^2 - 6 x + 9 + y^2 - 8 y + 16 = 4
x^2 - 6 x + y^2 - 8 y = -21