Write the equations in standard form. Label the center and the radius. Show work.
1. x^2+y^2+4x-8y+20=9
2. x^+y^2+8x+10y+41=49
3. x^2+y^2+10x-6y+34=16
4. x^2+y^2+4x+4y+8=16
The standard form of the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
With a center of (h, k) and a radius of r
1. x^2 + 4x + y^2 - 8y = -12
Complete the squares!
x^2 + 4x + 4 - 4 + y^2 - 8y + 16 - 16 = -12
By both adding and subtracting 4 and 16, the equation remains the same, but we have manipulated it so we can make perfect squares:
(x + 2)^2 + (y - 4)^2 - 4 - 16 = -12
(x + 2)^2 + (y - 4)^2 = 8
Center: (-2, 4)
Radius: √8 or 2√2
2. x^2 + 8x + y^2 + 10y = 8
x^2 + 8x + 16 - 16 + y^2 + 10y + 25 - 25 = 8
(x + 4)^2 + (y + 5)^2 - 16 - 25 = 8
(x + 4)^2 + (y + 5)^2 = 49
Center: (-4, -5)
Radius: √49 = 7
3. x^2 + 10x + y^2 - 6y = -18
x^2 + 10x + 25 - 25 + y^2 - 6y + 9 - 9 = -18
(x + 5)^2 + (y - 3)^2 = 16
Center: (-5, 3)
Radius: √16 = 4
4. x^2 + 4x + y^2 + 4y = 8
x^2 + 4x + 4 - 4 + y^2 + 4y + 4 - 4 = 8
(x + 2)^2 + (y + 2)^2 = 16
Center: (-2, -2)
Radius: 4
Hope this helps!!!
Comments
The standard form of the equation of a circle is:
(x - h)^2 + (y - k)^2 = r^2
With a center of (h, k) and a radius of r
1. x^2 + 4x + y^2 - 8y = -12
Complete the squares!
x^2 + 4x + 4 - 4 + y^2 - 8y + 16 - 16 = -12
By both adding and subtracting 4 and 16, the equation remains the same, but we have manipulated it so we can make perfect squares:
(x + 2)^2 + (y - 4)^2 - 4 - 16 = -12
(x + 2)^2 + (y - 4)^2 = 8
Center: (-2, 4)
Radius: √8 or 2√2
2. x^2 + 8x + y^2 + 10y = 8
x^2 + 8x + 16 - 16 + y^2 + 10y + 25 - 25 = 8
(x + 4)^2 + (y + 5)^2 - 16 - 25 = 8
(x + 4)^2 + (y + 5)^2 = 49
Center: (-4, -5)
Radius: √49 = 7
3. x^2 + 10x + y^2 - 6y = -18
x^2 + 10x + 25 - 25 + y^2 - 6y + 9 - 9 = -18
(x + 5)^2 + (y - 3)^2 = 16
Center: (-5, 3)
Radius: √16 = 4
4. x^2 + 4x + y^2 + 4y = 8
x^2 + 4x + 4 - 4 + y^2 + 4y + 4 - 4 = 8
(x + 2)^2 + (y + 2)^2 = 16
Center: (-2, -2)
Radius: 4
Hope this helps!!!