The coordinate of foci of the hyperbola `x^2/9-y^2/16=1` is

Options

A. `(+-4,0)`
B. `(+-5,0)`
C. `(+-6,0)`
D. `(+-3,0)`

Answer

To find the coordinates of the foci of the hyperbola given by the equation \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \), we will follow these steps:

### Step 1: Identify the values of \( a^2 \) and \( b^2 \)
The given equation of the hyperbola is in the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

From the equation:
- \( a^2 = 9 \)
- \( b^2 = 16 \)

### Step 2: Calculate \( a \) and \( b \)
To find \( a \) and \( b \), we take the square root of \( a^2 \) and \( b^2 \):
- \( a = \sqrt{9} = 3 \)
- \( b = \sqrt{16} = 4 \)

### Step 3: Calculate the value of \( c \)
For hyperbolas, the relationship between \( a \), \( b \), and \( c \) (the distance from the center to the foci) is given by the formula:
\[ c = \sqrt{a^2 + b^2} \]
Substituting the values of \( a^2 \) and \( b^2 \):
\[ c = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Determine the coordinates of the foci
For a hyperbola of the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the coordinates of the foci are given by:
\[ (\pm c, 0) \]
Thus, the coordinates of the foci are:
\[ (5, 0) \text{ and } (-5, 0) \]

### Final Answer
The coordinates of the foci of the hyperbola \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \) are \( (5, 0) \) and \( (-5, 0) \).

---