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 A)  (x - 21)^2 + (y - 19)^2 = 127

B) (x + 21)^2 + (y + 19)^2 = 849
C) (x + 21)^2 + (y + 19)^2 = 851
D) (x - 19)^2 + (y - 21)^2 = 2,209 .

The center of the circle is at the point

A) (-19, -21) 

B) (-21, -19) 

C) (19, 21) 

D) (21, 19) , 

and its radius is 

A) 127^(1/2)

B) 849^(1/2) 

C) 851^(1/2)

D) 47 units.

The general form of the equation of a circle that has the same radius as the above circle is 

A) x^2 + y^2 + 60x + 14y + 98 = 0 

B) x^2 + y^2 + 44x - 44y + 117 = 0 

C) x^2 + y^2 - 38x + 42y + 74 = 0 

D) x^2 + y^2 - 50x - 30y + 1 = 0  .

Answer 1

The common way of transforming the general form to the standard form is by using the completing the square method. But there is also a neat shortcut to knowing the h, k and r values right away.
The general formula is:
x²+y²+Dx+Ey+F = 0
x² + y² + (-2h)x + (-2k)y + (h²+k²-r²) = 0

Solving for h:
-2h = 42
h = -21

Solving for k:
-2k = 38
k = -19

Solving for r:
-47 = (-21)² + (-19)² - r²
r = √849

Standard form of given circle is:
(x+21)² + (y+19)² = 849

Answers:
1) B
2) B
3) B
4) D