To find the radius of the circle represented by the equation x² + y² – 8x – 6y – 21 = 0, we first need to rewrite this 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.
Starting with the given equation, we can rearrange it as follows:
x² – 8x + y² – 6y = 21
Next, we will complete the square for both the x and y terms.
Completing the Square for x:
Take the coefficient of x, which is -8, halve it to get -4, and then square it to get 16. We will add and subtract 16:
x² – 8x + 16 – 16
This simplifies to:
(x – 4)² – 16
Now the equation looks like this:
(x – 4)² – 16 + y² – 6y = 21
Completing the Square for y:
Take the coefficient of y, which is -6, halve it to get -3, and then square it to get 9. We will add and subtract 9:
y² – 6y + 9 – 9
This simplifies to:
(y – 3)² – 9
Now substitute this back into the equation:
(x – 4)² – 16 + (y – 3)² – 9 = 21
Combine like terms:
(x – 4)² + (y – 3)² – 25 = 21
Adding 25 to both sides gives:
(x – 4)² + (y – 3)² = 46
Now we see that the equation is in standard form, where:
- h = 4
- k = 3
- r² = 46
To find the radius r, we take the square root of r²:
r = √46
Thus, the radius of the circle is approximately:
r ≈ 6.78 units
In conclusion, the radius of the circle defined by the equation x² + y² – 8x – 6y – 21 = 0 is approximately 6.78 units.