To find the radius of the circle given by the equation x² + y² – 8x – 6y – 21 = 0, we first need to rewrite the equation in standard form. The standard form of a circle’s equation is:
(x – h)² + (y – k)² = r²
where (h, k) is the center of the circle and r is the radius.
Start by rearranging the given equation:
x² – 8x + y² – 6y = 21
Next, we need to complete the square for the x and y terms.
For the x terms:
- Take half of -8, square it: (-4)² = 16
- Add and subtract 16 in the equation.
For the y terms:
- Take half of -6, square it: (-3)² = 9
- Add and subtract 9 in the equation.
This gives us:
(x² – 8x + 16) + (y² – 6y + 9) = 21 + 16 + 9
Now simplify:
(x – 4)² + (y – 3)² = 46
From here, we see that the center of the circle is (4, 3) and the right side of the equation, 46, is equal to r².
To find the radius, we take the square root:
r = √46
Thus, the radius of the circle is approximately 6.78 units.