Understanding the Conic Section
To identify the conic section represented by the given equation, we need to first rearrange it into a standard form. The equation in question is:
x² + 4x + y² + 4y – 4 = 12
We will start by moving all terms to one side of the equation:
x² + 4x + y² + 4y – 16 = 0
Next, we will complete the square for both the x and y terms.
Completing the Square
For the x-terms:
- Take the coefficient of x, which is 4.
- Divide it by 2: 4 / 2 = 2
- Square this value: 2² = 4
- Add and subtract this square inside the equation.
For the y-terms:
- Similarly, the coefficient of y is 4.
- Divide it by 2: 4 / 2 = 2
- Square this value: 2² = 4
- Add and subtract this square inside the equation.
Now, our equation will look like this:
(x² + 4x + 4) + (y² + 4y + 4) – 16 = 0
Which simplifies to:
(x + 2)² + (y + 2)² – 16 = 0
Now, moving the constant to the other side, we get:
(x + 2)² + (y + 2)² = 16
Identifying the Conic Section
This equation is now in the standard form of a circle:
(x – h)² + (y – k)² = r²
Where (h, k) is the center of the circle and r is the radius. Here,
- The center is (-2, -2).
- The radius is √16, which is 4.
Conclusion
Thus, the conic section represented by the original equation, x² + 4x + y² + 4y – 4 = 12, is a circle centered at (-2, -2) with a radius of 4.