To find the radius of a circle from its equation, we first need to rewrite the given equation in the standard form of a circle’s equation. The standard form is:
(x – h)² + (y – k)² = r²
where (h, k) is the center of the circle and r is the radius.
Here, you have the equation:
x² + y² – 8x – 6y – 21 = 0
We will start by reorganizing this equation:
x² – 8x + y² – 6y = 21
Next, we complete the square for both the x and y terms.
Completing the square for x:
Take the coefficient of x, which is -8, divide it by 2 to get -4, and then square it to get 16.
Add and subtract 16:
(x² – 8x + 16) – 16
Completing the square for y:
Take the coefficient of y, which is -6, divide it by 2 to get -3, and then square it to get 9.
Add and subtract 9:
(y² – 6y + 9) – 9
Now substituting these completed squares back into the equation gives us:
(x – 4)² – 16 + (y – 3)² – 9 = 21
Simplifying this, we combine constants:
(x – 4)² + (y – 3)² – 25 = 21
This leads to:
(x – 4)² + (y – 3)² = 46
Now we can see that this is in the standard form of a circle’s equation, where:
- The center of the circle is at (4, 3)
- The right side of the equation, 46, represents r².
To find the radius, take the square root of 46:
r = √46
Thus, the radius of the circle is:
r ≈ 6.78
This means the radius of the circle described by the equation x² + y² – 8x – 6y – 21 = 0 is approximately 6.78 units.