Question

The general form of the equation of a circle is x2 + y2 + 42x + 38y − 47 = 0. The equation of this circle in standard form is (x - 21)^2 + (y - 19)^2 = 127 (x + 21)^2 + (y + 19)^2 = 849 (x + 21)^2 + (y + 19)^2 = 851 (x - 19)^2 + (y - 21)^2 = 2,209 . The center of the circle is at the point (-19, -21) (-21, -19) (19, 21) (21, 19) , and its radius is 127^(1/2) 849^(1/2) 851^(1/2) 47 units. The general form of the equation of a circle that has the same radius as the above circle is x^2 + y^2 + 60x + 14y + 98 = 0 x^2 + y^2 + 44x - 44y + 117 = 0 x^2 + y^2 - 38x + 42y + 74 = 0 x^2 + y^2 - 50x - 30y + 1 = 0 .

Answer 1

We have the following equation:
 x2 + y2 + 42x + 38y - 47 = 0
 We rewrite the equation:
 x2 + 42x + y2 + 38y - 47 = 0
 x2 + 42x + y2 + 38y = 47
 Rewriting we have:
 x2 + 42x + (42/2) ^ 2 + y2 + 38y + (38/2) ^ 2 = 47 + (42/2) ^ 2 + (38/2) ^ 2
 x2 + 42x + 441 + y2 + 38y + 361 = 47 + 441 + 361
 Rewriting we have:
 (x + 21) ^ 2 + (y + 19) ^ 2 = 849
 The center of the circle is:
 (x, y) = (-21, -19)
 The radio is:
 r = root (849)
 r = (849) ^ 2

 A circle of the same radius is given by:
 x ^ 2 + y ^ 2 - 50x - 30y + 1 = 0
 Let's check:
 x ^ 2 - 50x + y ^ 2 - 30y + 1 = 0
 x ^ 2 - 50x + y ^ 2 - 30y = - 1
 x ^ 2 - 50x + (-50/2) ^ 2 + y ^ 2 - 30y + (-30/2) ^ 2 = - 1 + (-30/2) ^ 2 + (-50/2) ^ 2
 x ^ 2 - 50x + (-50/2) ^ 2 + y ^ 2 - 30y + (-30/2) ^ 2 = - 1 + 225 + 625
 (x-25) ^ 2 + (y-15) ^ 2 = 849
 Answer:
 (x + 21) ^ 2 + (y + 19) ^ 2 = 849
 (x, y) = (-21, -19)
 r = (849) ^ 2
 x ^ 2 + y ^ 2 - 50x - 30y + 1 = 0