To find the center and the radius of a circle when given its equation in general form, you must first identify the equation in its standard form.
The standard form of a circle's equation is typically written as:
(x - h)² + (y - k)² = r²
where (h, k) represents the center of the circle and r is the radius.
The equation you have given seems to be in a non-standard format and could be interpreted as:
2x² + 2y² = 16
To convert this equation into standard form, we need to isolate x² and y² on the left-hand side and express the equation in terms of
(x - h)² and (y - k)².
Divide both sides of the equation by 2 to simplify the terms associated with x² and y²:
x² + y² = 8
This equation now resembles the standard form, where:
- h = 0 because there is no term subtracted from x, which implies x - 0 or simply x.
- k = 0 because there is no term subtracted from y, which implies y - 0 or simply y.
- r² = 8
Now we can see that the center of the circle (h, k) is (0, 0) because the equation does not have (x - h) or (y - k) but just x² and y².
To find the radius r, we take the square root of r²:
r = √8
To simplify the square root of 8, we can write it as:
r = √(4 * 2)
Since √4 is 2, we can simplify further:
r = 2√2
So, the radius of the circle is 2√2.
In summary, the center of the circle is (0, 0) and the radius is 2√2.
