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
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
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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
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 !
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
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
To transform an equation into center-radius form, you need to complete the square for both the x-terms and y-terms separately. Here's how to solve each of the given equations:
1. x^2 + y^2 + 8x - 14y + 1 = 0
To complete the square for the x-terms, you need to add (8/2)^2 = 16 to both sides of the equation:
x^2 + 8x + 16 + y^2 - 14y + 1 = 16
Now, rewrite the equation by grouping the x-terms and y-terms separately:
(x^2 + 8x + 16) + (y^2 - 14y) + 1 = 16
For the x-terms, you can factor it as a perfect square: (x + 4)^2 = x^2 + 8x + 16.
For the y-terms, complete the square by adding (14/2)^2 = 49 to both sides of the equation:
(x + 4)^2 + (y^2 - 14y + 49) + 1 - 16 = 0
Simplify the equation:
(x + 4)^2 + (y - 7)^2 - 16 = 0
So the center is (-4, 7) and the radius is √16 = 4.
2. y^2 + x^2 - 2x - 12y - 12 = 0
To complete the square for the x-terms, you need to add (-2/2)^2 = 1 to both sides of the equation:
x^2 - 2x + 1 + y^2 - 12y - 12 = 1
Now, rewrite the equation by grouping the x-terms and y-terms separately:
(x^2 - 2x + 1) + (y^2 - 12y) - 11 = 1
For the x-terms, you can factor it as a perfect square: (x - 1)^2 = x^2 - 2x + 1.
For the y-terms, complete the square by adding (-12/2)^2 = 36 to both sides of the equation:
(x - 1)^2 + (y^2 - 12y + 36) + 1 - 11 = 0
Simplify the equation:
(x - 1)^2 + (y - 6)^2 - 10 = 0
So the center is (1, 6) and the radius is √10.
3. x^2 + y^2 + 4x - 5 = 0
To complete the square for the x-terms, you need to add (4/2)^2 = 4 to both sides of the equation:
x^2 + 4x + 4 + y^2 - 5 = 4
Now, rewrite the equation by grouping the x-terms and y-terms separately:
(x^2 + 4x + 4) + y^2 - 1 = 4
For the x-terms, you can factor it as a perfect square: (x + 2)^2 = x^2 + 4x + 4.
Simplify the equation:
(x + 2)^2 + y^2 - 1 - 4 = 0
(x + 2)^2 + y^2 - 5 = 0
So the center is (-2, 0) and the radius is √5.
4. y^2 + x^2 - 6y + 5 = 0
To complete the square for the x-terms, you need to add (0/2)^2 = 0 to both sides of the equation:
x^2 + y^2 - 6y + 5 = 0
Now, rewrite the equation by grouping the x-terms and y-terms separately:
(x^2) + (y^2 - 6y) + 5 = 0
For the x-terms, no need to complete the square as it is already in the perfect square form: x^2.
For the y-terms, complete the square by adding (-6/2)^2 = 9 to both sides of the equation:
(x^2) + (y^2 - 6y + 9) + 5 - 9 = 0
Simplify the equation:
(x^2) + (y - 3)^2 - 4 = 0
So the center is (0, 3) and the radius is √4 = 2.
5. x^2 + y^2 - 256 = 0
To complete the square for the x-terms, you need to add (0/2)^2 = 0 to both sides of the equation:
x^2 + y^2 - 256 + 0 = 0
Now, rewrite the equation by grouping the x-terms and y-terms separately:
(x^2) + (y^2) - 256 = 0
For both the x-terms and y-terms, no need to complete the square as they are already in the perfect square form.
Simplify the equation:
(x^2) + (y^2) = 256
So the center is (0, 0) and the radius is √256 = 16.