From the point `P(3, 4)` pair of tangents PA and PB are drawn to the ellipse `(x^(2))/(16)+(y^(2))/(9)=1`. If AB intersects y - axis at C and x - axis at D, then OC. OD is equal to (where O is the origin)

Options

A. 9
B. 18
C. 8
D. 12

Answer

To solve the problem, we need to find the product of the distances from the origin \( O \) to the points \( C \) and \( D \) where the chord of contact \( AB \) intersects the y-axis and x-axis respectively.

### Step-by-Step Solution:

1. **Identify the equation of the ellipse**:
The given ellipse is:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
Here, \( a^2 = 16 \) and \( b^2 = 9 \).

2. **Find the chord of contact from point \( P(3, 4) \)**:
The equation of the chord of contact for an ellipse from an external point \( (x_1, y_1) \) is given by:
\[ \frac{x_1 x}{a^2} + \frac{y_1 y}{b^2} - 1 = 0 \]
Substituting \( x_1 = 3 \), \( y_1 = 4 \), \( a^2 = 16 \), and \( b^2 = 9 \):
\[ \frac{3x}{16} + \frac{4y}{9} - 1 = 0 \]

3. **Rearranging the equation**:
Rearranging gives:
\[ 3x + \frac{64y}{9} = 16 \]
To eliminate the fraction, multiply through by 9:
\[ 27x + 64y = 144 \]

4. **Finding the intersection with the y-axis (point C)**:
To find point \( C \), set \( x = 0 \):
\[ 27(0) + 64y = 144 \implies 64y = 144 \implies y = \frac{144}{64} = \frac{9}{4} \]
Thus, the coordinates of point \( C \) are \( (0, \frac{9}{4}) \).

5. **Finding the intersection with the x-axis (point D)**:
To find point \( D \), set \( y = 0 \):
\[ 27x + 64(0) = 144 \implies 27x = 144 \implies x = \frac{144}{27} = \frac{16}{3} \]
Thus, the coordinates of point \( D \) are \( (\frac{16}{3}, 0) \).

6. **Calculating distances OC and OD**:
- Distance \( OC \) (from origin \( O(0, 0) \) to \( C(0, \frac{9}{4}) \)):
\[ OC = \frac{9}{4} \]
- Distance \( OD \) (from origin \( O(0, 0) \) to \( D(\frac{16}{3}, 0) \)):
\[ OD = \frac{16}{3} \]

7. **Finding the product \( OC \cdot OD \)**:
\[ OC \cdot OD = \left(\frac{9}{4}\right) \cdot \left(\frac{16}{3}\right) = \frac{9 \cdot 16}{4 \cdot 3} = \frac{144}{12} = 12 \]

### Final Answer:
Thus, the value of \( OC \cdot OD \) is \( 12 \).