Rewriting the Equation in Standard Form
The given equation is:
x² + y² + 8x + 22y + 37 = 0
To express this equation in standard form, we need to reorganize the expression by completing the square for both the x and y terms.
Step 1: Group the x and y terms
First, we rearrange the equation:
x² + 8x + y² + 22y + 37 = 0
Step 2: Completing the square for x terms
For the x terms, we take:
x² + 8x
To complete the square, we take half of the coefficient of x (which is 8) and square it:
(8/2)² = 16
Now we can express the x terms as:
(x + 4)² – 16 (since we added 16 and need to subtract it to keep the equation balanced)
Step 3: Completing the square for y terms
Now, we apply the same technique to the y terms:
y² + 22y
Half of 22 is 11, and squaring it gives:
(11)² = 121
Thus, we rewrite the y terms as:
(y + 11)² – 121
Step 4: Substituting back into the equation
Substituting the completed squares back into the equation, we get:
(x + 4)² – 16 + (y + 11)² – 121 + 37 = 0
Simplifying this:
(x + 4)² + (y + 11)² – 100 = 0
Final Step: Isolating the standard form
To express the equation in standard form, we isolate the radius term on one side:
(x + 4)² + (y + 11)² = 100
This equation is now in the standard form of a circle, which is:
(x – h)² + (y – k)² = r²,
where (h, k) is the center of the circle and r is the radius.
Conclusion
Thus, the equation x² + y² + 8x + 22y + 37 = 0 in standard form is:
(x + 4)² + (y + 11)² = 100
with a center at (-4, -11) and a radius of 10.