Question

Using the following equation, find the center and radius of the circle. You must show all work and calculations to receive credit. x2 - 4x + y2 + 8y = -4

Answer 1

The center-radius form of the circle equation is in the format (x – h)^2 + (y – k)^2 = r^2, with the center being at the point (h, k) and the radius being "r". We do as follows:

x^2 − 4x + y^2 + 8y = −4

(x^2 − 4x) + (y^2 + 8y) = −4

(x^2 − 4x + 4) + (y^2 + 8y + 16) = −4 +4 + 16

(x - 2)^2 + ( y + 4)^2 = 16 = 4^2

Center: (-2 , 4 )

radius: 4

Answer 2

The equation of the circle: $(x-a)^2+(y-b)^2=r^2$

where:
(a; b) - coordinates of the center
r - a radius

Use: $(a\pm b)^2=a^2\pm2ab+b^2\ \ \ \ (*)$

We have:

$x^2-4x+y^2+8y=-4\\\\x^2-2x\cdot2+y^2+2y\cdot4=-4\ \ \ |add\ 2^2\ and\ 4^2\ to\ both\ sides\\\\\underbrace{x^2-2x\cdot2+2^2}_{(*)}+\underbrace{y^2+2y\cdot4+4^2}_{(*)}=-4+2^2+4^2\\\\(x-2)^2+(y+4)^2=-4+4+16\\\\(x+2)^2+(y-4)^2=16\\\\(x+2)^2+(y-4)^2=4^2$

Answer:
The center: (-2; 4)
The radius: r = 4