what is radius of 4x^2 + 4y^2 + 8x - 92 = 8y?

Question

center= (-1,1)

Answer 1

If the center is (-1,1), the the circle looks like

(x - -1)^2 + (y-1)^2 = r^2 or
(x + 1)^2 + (y-1)^2 = r^2.

4x^2 + 4y^2 + 8x - 92 = 8y, divide everything by 4

x^2 + y^2 + 2x - 23 = 2y, rearrange as

x^2 + 2x + y^2 -2y = 23, note sign changes, complete the square
(x^2 + 2x + 1) + (y^2 -2y +1) = 23 + 1 + 1, note, we just added 2 to both sides, so did not change anything

(x+1)^2 + (y-1)^2 = 5^2, radius is 5.

Answer 2

4x² + 4y² + 8x - 92 = 8y
=> x² + y² + 2x - 23 = 2y
=> x² +2x + y² -2y - 23 = 0
=> x² +2x + 1 + y² -2y + 1 - 23 - 2 = 0
=> (x + 1)² + (y - 1)² = 25
Since equation of circle is of form (x - a)² + (y - b)² = r²
Radius = √(25)
= 5

Answer 3

4x^2 + 4y^2 + 8x - 92 = 8y
4x^2 + 4y^2 + 8x - 8y = 92
4x^2 + 8x + 4y^2 - 8y = 92
x^2 + 2x + y^2 - 2y = 23
(x^2 + 2x) + (y^2 - 2y) = 23
(x^2 + 2x + 1 - 1) + (y^2 - 2y + 1 - 1) = 23
((x + 1)^2 - 1) + ((y - 1)^2 - 1) = 23
(x + 1)^2 - 1 + (y - 1)^2 - 1 = 23
(x + 1)^2 + (y - 1)^2 - 2 = 23
(x + 1)^2 + (y - 1)^2 = 25

now using

(x - h)^2 + (y - k)^2 = r^2

ANS : radius is 5

Answer 4

x² + y² + 2x - 2y - 23 = 0
x² + y² + 2gx + 2fy + c = 0
Centre(- g ,- f)
c = - 23
r ² = g ² + f ² - c
2 g = 2
g = 1
2 f = - 2
f = - 1
Centre(-1 , 1)
r ² = 1 + 1 + 23
r ² = 25
r = 5