The hyberbola `(x^(2)+4x+4)/(25)-(y^(2)-6x+9)/(16)=1` is centered at which of the following points ?

Options

A. (-4, -9)
B. (-2, 3)
C. (2, -3)
D. (5, 4)

Explanation

To find the center of the hyperbola given by the equation

\[ \frac{x^2 + 4x + 4}{25} - \frac{y^2 - 6x + 9}{16} = 1, \]

we will follow these steps:

### Step 1: Rewrite the equation in standard form

The standard form of a hyperbola is

\[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1, \]

where \((h, k)\) is the center of the hyperbola.

### Step 2: Complete the square for the \(x\) terms

The \(x\) terms in the equation are \(x^2 + 4x + 4\). We can complete the square:

\[ x^2 + 4x + 4 = (x + 2)^2. \]

### Step 3: Rewrite the equation with the completed square

Substituting the completed square back into the equation gives:

\[ \frac{(x + 2)^2}{25} - \frac{y^2 - 6x + 9}{16} = 1. \]

### Step 4: Complete the square for the \(y\) terms

Now, we need to simplify the \(y\) terms, which are \(y^2 - 6x + 9\). We can rewrite it as:

\[ y^2 - 6x + 9 = (y - 3)^2. \]

### Step 5: Substitute back into the equation

Now, substituting this back into the equation gives:

\[ \frac{(x + 2)^2}{25} - \frac{(y - 3)^2}{16} = 1. \]

### Step 6: Identify the center of the hyperbola

From the standard form, we can see that \(h = -2\) and \(k = 3\). Therefore, the center of the hyperbola is:

\[ (-2, 3). \]

### Conclusion

Thus, the center of the hyperbola is at the point \((-2, 3)\), which corresponds to option **b**.

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