The general equation of a circle.
The general equation of a circle is (x-h)² + (y-k)² = r²
Where,
- (h,k) = center of circle
- r = radius
Here we are given the equation (x+4)² + (y+5)² = 64 and given this we are asked to find the diameter as well as the coordinates of the center.
Finding the coordinates of the center and diameter.
Finding the center
If (h,k) = center, then the coordinates of the center are the given values of h and k but negative. This is because the values are being subtracted from x and y. (x-h)² + (y-k)² = r²
This means that, (x+4)² + (y+5)² = 64 is the same as saying (x-(-4))² + (y-(-5))² = 64 based off of the general equation of a circle.
Notice how the plugged in values are negative rather than positive.
In conclusion, we can say that the center is at (-4,-5) (see attached image below to check answer)
In the equation, 64 takes the place of r²
So r² = 64
Taking the square root of both sides we get r = 8
So the radius is 8.
To convert to diameter we multiply by 2
So diameter = 8 * 2 = 16
In conclusion, a circle with the equation (x+4)² + (y + 5)² = 64, has a center at (-4,-5) and a diameter of 16