Question

what is are the coordinates of the center and length of the radius of the circle whose equation is x^2 + 6x + y^2 -4y =23

Answer 1

The general equation of a circle is:
`(x-a)^2+(y-b)^2=r^2`, where
`(a,b)` is the center of the circle and
`r` is its radius. We'll put the equation of the statement in the general form:


`x^2+6x+y^2-4y=23\\\\ x^2+6x+\underline{9}+y^2-4y+\underline{4}=23+\underline{9}+\underline{4}\\\\ (x+3)^2+(y-2)^2=36\\\\ (x+3)^2+(y-2)^2=6^2`

Then, by comparing the general equation of the circle and the equation above, we have:


`\boxed{\text{Center:}~(-3,2)}\\\\ \boxed{\text{Radius:}~r=6~u.c.}`

Answer 2

The coordinates (x,y) are -3,2 while the radius is 6cm

Explanation:

The general form of the equation of a circle may be given as

(x - a)² + (y - b)² = c²

where c is the radius, a and b are the coordinates of the center of the circle along the x and y axis.

Given the circle

x^2 + 6x + y^2 - 4y =23

To make it similar to the general form

x^2 + 6x +9 + y^2 -4y + 4 =23 + 9 + 4

Factorizing

(x + 3)² + (y - 2)² = 36

Comparing with the general form,

c² = 36

c = √36

= 6 cm ( this is the radius)

a = -3, b = 2