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 step by step, we will follow the process of finding the equations of the tangents from the point P(3, 4) to the given ellipse and then determine the intercepts on the axes.

### Step 1: Identify the equation of the ellipse
The given equation of the ellipse is:
\[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \]
Here, \(a^2 = 16\) and \(b^2 = 9\), so \(a = 4\) and \(b = 3\).

### Step 2: Use the formula for the equation of the chord of contact
The equation of the chord of contact from a point \(P(x_1, y_1)\) to the ellipse is given by:
\[ \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \]
Substituting \(x_1 = 3\) and \(y_1 = 4\):
\[ \frac{3x}{16} + \frac{4y}{9} = 1 \]

### Step 3: Rearranging the equation
To make it easier to find intercepts, we can rearrange the equation:
\[ 3x + \frac{64y}{9} = 16 \]
Multiplying through by 9 to eliminate the fraction:
\[ 27x + 64y = 144 \]

### Step 4: Find the y-intercept (C)
To find the y-intercept (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 \((0, \frac{9}{4})\).

### Step 5: Find the x-intercept (D)
To find the x-intercept (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 \((\frac{16}{3}, 0)\).

### Step 6: Calculate OC and OD
Now, we can find the distances \(OC\) and \(OD\):
- \(OC\) is the distance from the origin \(O(0, 0)\) to \(C(0, \frac{9}{4})\):
\[ OC = \frac{9}{4} \]
- \(OD\) is the distance from the origin \(O(0, 0)\) to \(D(\frac{16}{3}, 0)\):
\[ OD = \frac{16}{3} \]

### Step 7: Calculate the product \(OC \cdot OD\)
Now we calculate \(OC \cdot OD\):
\[ OC \cdot OD = \left(\frac{9}{4}\right) \cdot \left(\frac{16}{3}\right) = \frac{9 \cdot 16}{4 \cdot 3} = \frac{144}{12} = 12 \]

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

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