If P is a point on the hyperbola ` 16x^(2) - 9y^(2) = 144` whose foci are ` S_(1)" and " S_(2) " then : " | S_(1) P - S_(2) P | = `

Options

A. 4
B. 6
C. 8
D. 12

Answer

To solve the problem, we need to find the value of \( | S_1P - S_2P | \) for a point \( P \) on the hyperbola given by the equation:

\[ 16x^2 - 9y^2 = 144 \]

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

First, we need to rewrite the equation of the hyperbola in standard form. We do this by dividing the entire equation by 144:

\[ \frac{16x^2}{144} - \frac{9y^2}{144} = 1 \]

This simplifies to:

\[ \frac{x^2}{9} - \frac{y^2}{16} = 1 \]

### Step 2: Identify the values of \( a \) and \( b \)

From the standard form of the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), we can identify:

\[ a^2 = 9 \quad \Rightarrow \quad a = \sqrt{9} = 3 \]
\[ b^2 = 16 \quad \Rightarrow \quad b = \sqrt{16} = 4 \]

### Step 3: Calculate the distance between the foci

The foci of a hyperbola are located at \( (c, 0) \) and \( (-c, 0) \), where \( c \) is given by the formula:

\[ c = \sqrt{a^2 + b^2} \]

Calculating \( c \):

\[ c = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Thus, the foci \( S_1 \) and \( S_2 \) are located at \( (5, 0) \) and \( (-5, 0) \).

### Step 4: Use the property of hyperbolas

For any point \( P \) on the hyperbola, the property of hyperbolas states that:

\[ | S_1P - S_2P | = 2a \]

### Step 5: Calculate \( | S_1P - S_2P | \)

Now substituting the value of \( a \):

\[ | S_1P - S_2P | = 2 \times a = 2 \times 3 = 6 \]

### Final Answer

Thus, the value of \( | S_1P - S_2P | \) is:

\[ \boxed{6} \]

---