Let `veca,vecb` be the position vectors of points A and B with respect to and `|veca|=a,|vecb|=b`. The points Cand D divide AB internally and externally in the ratio 2: 3 respectively. If `vec(OC)andvec(OD)` are perpendicular, then

Options

A. `9a^(2)=4b^(2)`
B. `4a^(2)=9b^(2)`
C. 9a=4b
D. 4a=9b

Answer

To solve the problem, we need to find the relationship between the magnitudes of the position vectors of points A and B given that points C and D divide the line segment AB in certain ratios and that the vectors OC and OD are perpendicular.

### Step-by-Step Solution:

1. **Identify the Position Vectors**:
Let the position vectors of points A and B be represented as:
\[ \vec{a} \quad \text{and} \quad \vec{b} \]
Given that \( |\vec{a}| = a \) and \( |\vec{b}| = b \).

2. **Find the Position Vector of Point C**:
Point C divides the line segment AB internally in the ratio 2:3. The formula for the position vector of a point dividing a line segment internally in the ratio \( m:n \) is:
\[ \vec{C} = \frac{m\vec{b} + n\vec{a}}{m+n} \]
For our case, \( m = 2 \) and \( n = 3 \):
\[ \vec{C} = \frac{2\vec{b} + 3\vec{a}}{2 + 3} = \frac{2\vec{b} + 3\vec{a}}{5} \]

3. **Find the Position Vector of Point D**:
Point D divides the line segment AB externally in the ratio 2:3. The formula for the position vector of a point dividing a line segment externally in the ratio \( m:n \) is:
\[ \vec{D} = \frac{m\vec{b} - n\vec{a}}{m-n} \]
For our case, \( m = 2 \) and \( n = 3 \):
\[ \vec{D} = \frac{2\vec{b} - 3\vec{a}}{2 - 3} = \frac{2\vec{b} - 3\vec{a}}{-1} = 3\vec{a} - 2\vec{b} \]

4. **Condition for Perpendicularity**:
The vectors \( \vec{C} \) and \( \vec{D} \) are perpendicular, which means:
\[ \vec{C} \cdot \vec{D} = 0 \]
Substituting the expressions for \( \vec{C} \) and \( \vec{D} \):
\[ \left(\frac{2\vec{b} + 3\vec{a}}{5}\right) \cdot (3\vec{a} - 2\vec{b}) = 0 \]

5. **Expanding the Dot Product**:
Expanding the dot product:
\[ \frac{1}{5} \left( (2\vec{b}) \cdot (3\vec{a}) + (3\vec{a}) \cdot (3\vec{a}) - (2\vec{b}) \cdot (2\vec{b}) \right) = 0 \]
Simplifying:
\[ \frac{1}{5} \left( 6 \vec{a} \cdot \vec{b} + 9 \vec{a} \cdot \vec{a} - 4 \vec{b} \cdot \vec{b} \right) = 0 \]
This leads to:
\[ 6 \vec{a} \cdot \vec{b} + 9 |\vec{a}|^2 - 4 |\vec{b}|^2 = 0 \]

6. **Substituting Magnitudes**:
Using \( |\vec{a}| = a \) and \( |\vec{b}| = b \):
\[ 6 (\vec{a} \cdot \vec{b}) + 9a^2 - 4b^2 = 0 \]

7. **Rearranging the Equation**:
Rearranging gives:
\[ 9a^2 = 4b^2 - 6 (\vec{a} \cdot \vec{b}) \]

### Final Result:
The final equation we derived is:
\[ 9a^2 = 4b^2 \]