The curve represented by the equation `4x^(2)+16y^(2)-24x-24x-32y-12=0`is

Options

A. a parabola
B. a pair of stright lines
C. an ellipse with eccentricity `(1)/(2)`
D. an ellipse with eccentricity `(sqrt(3))/(2)`

Explanation

To solve the equation \(4x^2 + 16y^2 - 24x - 32y - 12 = 0\) and determine the type of curve it represents, we will follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the equation to group the \(x\) and \(y\) terms together:
\[ 4x^2 - 24x + 16y^2 - 32y = 12 \]

**Hint:** Look for terms that can be grouped together for completing the square.

### Step 2: Completing the Square for \(x\)
For the \(x\) terms \(4x^2 - 24x\):
1. Factor out the coefficient of \(x^2\):
\[ 4(x^2 - 6x) \]
2. To complete the square, take half of the coefficient of \(x\) (which is \(-6\)), square it, and add/subtract it inside the bracket:
\[ 4\left(x^2 - 6x + 9 - 9\right) = 4\left((x - 3)^2 - 9\right) = 4(x - 3)^2 - 36 \]

**Hint:** Remember to balance the equation by adjusting the constant term when completing the square.

### Step 3: Completing the Square for \(y\)
For the \(y\) terms \(16y^2 - 32y\):
1. Factor out the coefficient of \(y^2\):
\[ 16(y^2 - 2y) \]
2. Complete the square:
\[ 16\left(y^2 - 2y + 1 - 1\right) = 16\left((y - 1)^2 - 1\right) = 16(y - 1)^2 - 16 \]

**Hint:** Use the same method of completing the square as with the \(x\) terms.

### Step 4: Substitute Back into the Equation
Now substitute the completed squares back into the equation:
\[ 4(x - 3)^2 - 36 + 16(y - 1)^2 - 16 = 12 \]
Combine the constant terms:
\[ 4(x - 3)^2 + 16(y - 1)^2 - 52 = 12 \]
\[ 4(x - 3)^2 + 16(y - 1)^2 = 64 \]

**Hint:** Always ensure that the equation remains balanced after substituting back.

### Step 5: Divide by 64
To express the equation in standard form, divide the entire equation by 64:
\[ \frac{4(x - 3)^2}{64} + \frac{16(y - 1)^2}{64} = 1 \]
This simplifies to:
\[ \frac{(x - 3)^2}{16} + \frac{(y - 1)^2}{4} = 1 \]

**Hint:** The standard form of an ellipse is \(\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1\).

### Step 6: Identify the Curve
From the standard form \(\frac{(x - 3)^2}{16} + \frac{(y - 1)^2}{4} = 1\), we can see that this is the equation of an ellipse centered at \((3, 1)\) with semi-major axis \(a = 4\) and semi-minor axis \(b = 2\).

**Conclusion:** The curve represented by the equation \(4x^2 + 16y^2 - 24x - 32y - 12 = 0\) is an **ellipse**.