$$ 4 ( x ^ { 2 } - 2 x ) + ( y ^ { 2 } + 4 y ) = 8 $$

Question

UpStudy AI · Accepted Answer

Certainly! Let's analyze and manipulate the given equation step by step.

**Given Equation:**
$$
4(x^{2} - 2x) + (y^{2} + 4y) = 8
$$

**Objective:**
- Rewrite the equation in standard form by completing the squares.
- Identify the type of conic section represented by the equation.

### Step 1: Complete the Square for the $$ x $$-Terms

Start with the $$ x $$-terms:
$$
4(x^{2} - 2x)
$$

1. **Factor out the coefficient of $$ x^2 $$:**
   $$
   4(x^{2} - 2x) = 4\left[ x^{2} - 2x \right]
   $$

2. **Complete the square inside the parentheses:**
   $$
   x^{2} - 2x = \left( x^{2} - 2x + 1 \right) - 1 = (x - 1)^2 - 1
   $$

3. **Multiply back by 4:**
   $$
   4\left[ (x - 1)^2 - 1 \right] = 4(x - 1)^2 - 4
   $$

### Step 2: Complete the Square for the $$ y $$-Terms

Now, address the $$ y $$-terms:
$$
y^{2} + 4y
$$

1. **Complete the square:**
   $$
   y^{2} + 4y = \left( y^{2} + 4y + 4 \right) - 4 = (y + 2)^2 - 4
   $$

### Step 3: Substitute Back into the Original Equation

Replace the completed squares into the original equation:
$$
4(x - 1)^2 - 4 + (y + 2)^2 - 4 = 8
$$

Combine like terms:
$$
4(x - 1)^2 + (y + 2)^2 - 8 = 8
$$

Add 8 to both sides:
$$
4(x - 1)^2 + (y + 2)^2 = 16
$$

### Step 4: Divide to Obtain the Standard Form

Divide every term by 16 to normalize the equation:
$$
\frac{4(x - 1)^2}{16} + \frac{(y + 2)^2}{16} = \frac{16}{16}
$$
$$
\frac{(x - 1)^2}{4} + \frac{(y + 2)^2}{16} = 1
$$

### Step 5: Identify the Conic Section

The standard form of the equation is:
$$
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1
$$

Comparing, we have:
- $$ h = 1 $$
- $$ k = -2 $$
- $$ a^2 = 4 $$ ⇒ $$ a = 2 $$
- $$ b^2 = 16 $$ ⇒ $$ b = 4 $$

**Conclusion:**
- The equation represents an **ellipse**.
- **Center** of the ellipse: $$ (1, -2) $$.
- **Semi-major axis** (along the $$ y $$-axis): 4 units.
- **Semi-minor axis** (along the $$ x $$-axis): 2 units.

**Graphical Representation:**

![Ellipse](https://i.imgur.com/ellipse.png)

*Note: The image is a placeholder. In practice, you would plot the ellipse centered at (1, -2) with the specified axes lengths.*

### Summary

By completing the squares for both $$ x $$ and $$ y $$, the original equation is transformed into the standard form of an ellipse:
$$
\frac{(x - 1)^2}{4} + \frac{(y + 2)^2}{16} = 1
$$
This reveals the geometric properties of the ellipse, including its center and axis lengths.
The equation simplifies to an ellipse centered at (1, -2) with a semi-major axis of 4 units along the y-axis and a semi-minor axis of 2 units along the x-axis.

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