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

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:**  *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.