To transform general form to a standard form, you can use completing the square method or you can convert this equation x² + y² + Dx + Ey + F = 0 to x² + y² + (-2h)x + (-2k)y + (h²+k²-r²) = 0
Given: x2 + y2 + 42x + 38y − 47 = 0
Required: Find the values of h, k, and r using this formula x² + y² + (-2h)x + (-2k)y + (h²+k²-r²) = 0
Solution:
-2h = 42
h = 21
-2k = 38
k = 19
-47 = h^2 + k^2 - r^2
-47 = 21^2 + 19^2 - r^2
r^2 = 849
r = √849
Answers:
1.) Standard form : (x+21)² + (y+19)² = 849
2.) Center of the circle : (h,k) = (21,19)
3.) Radius of the circle : r = √849
4.) Equation of a circle that has the same radius as the above circle is : x^2 + y^2 - 50x - 30y + 1 = 0
You can solve the last one through trial and error. Its h - 25; k - 15.
1 = h^2 + k^2 - r^2
1 = -25^2 + -15^2 - r^2
r^2 = 849
r = √849 : which has the same radius as the given equation.
