Find the center and radius of the given circle.
1. x^2 + y^2 + 10y + 21 = 0
To find the center and radius of the given circle, we need to rewrite the equation in the standard form: (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the coordinates of the center and r represents the radius.
Let's start with the given equation:
x^2 + y^2 + 10y + 21 = 0
First, we complete the square for the y terms. To do this, we need to add and subtract a constant that completes the square. In this case, the constant we want to add is (10/2)^2 = 25.
x^2 + y^2 + 10y + 21 + 25 = 25
Simplifying the equation:
x^2 + y^2 + 10y + 46 = 25
Now, we can rewrite the equation as:
x^2 + (y^2 + 10y + 25) + 46 - 25 = 0
Simplifying further:
x^2 + (y + 5)^2 + 21 = 0
Comparing this to the standard form, we can see that the center is (-h, -k) and the radius is √r^2.
From the equation, we can determine that the center of the circle is at (-0, -5), which simplifies to (0, -5). And the radius is √21.
Therefore, the center of the given circle is (0, -5) and the radius is √21.