Question

Match the pairs of equations that represent concentric circles.

Answer 1

if the answer is correct it should look like this =) you welcome.

Answer 2

3x^2 + 3y^2 + 12x − 6y − 21 = 0 => x^2 + y^2 + 4x - 2y - 7 = 0 => x^2 + 4x + 4 + y^2 - 2y + 1 = 12 => (x + 2)^2 + (y - 1)^2 = 12 => centre is (-2, 1)

5x^2 + 5y^2 − 10x + 40y − 75 = 0 => x^2 + y^2 - 2x + 8y - 15 = 0 => x^2 - 2x + 1 + y^2 + 8y + 16 = 32 => (x - 1)^2 + (y + 4)^2 = 32 => centre is (1, -4)

5x^2 + 5y^2 − 30x + 20y − 10 = 0 => x^2 + y^2 - 6x + 4y - 2 = 0 => x^2 - 6x + 9 + y^2 + 4y + 4 = 15 => (x - 3)^2 + (y + 2)^2 = 15 => centre is (3, -2)

4x^2 + 4y^2 + 16x − 8y − 308 = 0 => x^2 + y^2 + 4x - 2y - 77 = 0 => x^2 + 4x + 4 + y^2 - 2y + 1 = 82 => (x + 2)^2 + (y - 1)^2 = 82 => centre is (-2, 1)
 
x^2 + y^2 − 12x − 8y − 100 = 0 => x^2 - 12x + 36 + y^2 - 8y + 16 = 152 => (x - 6)^2 + (y - 4)^2 = 152 => centre is (6, 4)

2x^2 + 2y^2 − 8x + 12y − 40 = 0 => x^2 + y^2 - 4x + 6y - 20 = 0 => x^2 - 4x + 4 + y^2 + 6y + 9 = 33 => (x - 2)^2 + (y + 3)^2 = 33 => centre is (2, -3)
 
4x^2 + 4y^2 − 16x + 24y − 28 = 0 => x^2 + y^2 - 4x + 6y - 7 = 0 => x^2 - 4x + 4 + y^2 + 6y + 9 = 20 => (x - 2)^2 + (y + 3)^2 = 20 => centre is (2, -3)

3x^2 + 3y^2 − 18x + 12y − 81 = 0 => x^2 + y^2 - 6x + 4y - 27 = 0 => x^2 - 6x + 9 + y^2 + 4y + 4 = 40 => (x - 3)^2 + (y + 2)^2 = 40 => centre is (3, -2)

x^2 + y^2 − 2x + 8y − 13 = 0 => x^2 - 2x + 1 + y^2 + 8y + 16 = 30 => (x - 1)^2 + (y + 4)^2 = 30 => centre = (1, -4)
 
x^2 + y^2 + 24x + 30y + 17 = 0 => x^2 + 24x + 144 + y^2 + 30y + 225 = 352 => (x + 12)^2 + (y + 15)^2 = 352 => center is (-12, -15)

Therefore, 3x^2 + 3y^2 + 12x − 6y − 21 = 0 and 4x^2 + 4y^2 + 16x − 8y − 308 = 0 are concentric.
 5x^2 + 5y^2 − 10x + 40y − 75 = 0 and x^2 + y^2 − 2x + 8y − 13 = 0 are concentric.
 5x^2 + 5y^2 − 30x + 20y − 10 = 0 and 3x^2 + 3y^2 − 18x + 12y − 81 = 0 are concentric.
 2x^2 + 2y^2 − 8x + 12y − 40 = 0 and 4x^2 + 4y^2 − 16x + 24y − 28 = 0 are concentric.