In a `Delta ABC ` the equation of the side BC is `2x-y =3` and its circumcentre and orthocentre are `(2,4) and (1,2) ` respetively .
`sin B.sin C=`

Options

A. `9/(2sqrt(61))`
B. `9/(4sqrt(61))`
C. `9/(sqrt(61))`
D. `9/(5sqrt(61))`

Answer

To solve the problem, we need to find the value of \( \sin B \cdot \sin C \) given the equation of side \( BC \) and the coordinates of the circumcenter and orthocenter.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The equation of side \( BC \) is \( 2x - y = 3 \).
- The circumcenter \( P \) is at \( (2, 4) \).
- The orthocenter \( O \) is at \( (1, 2) \).

2. **Find the Lengths \( OP \) and \( OD \):**
- The distance \( OP \) can be calculated using the distance formula:
\[ OP^2 = (2 - 1)^2 + (4 - 2)^2 = 1^2 + 2^2 = 1 + 4 = 5 \]
- Thus, \( OP = \sqrt{5} \).

3. **Use the Relationship Between Circumradius and Angles:**
- The relationship for circumradius \( R \) is given by:
\[ OP^2 = R^2(1 - \cos A) \]
- Since \( OP^2 = 5 \), we can express this as:
\[ 5 = R^2(1 - \cos A) \]

4. **Express \( R \cos A \):**
- From the circumradius formula, we have:
\[ R \cos A = \frac{2}{\sqrt{5}} \]

5. **Find \( R \cos B \cos C \):**
- The length \( OD \) can be expressed as:
\[ OD = 2R \cos B \cos C \]
- We need to find \( OD \) using the information given.

6. **Set Up the Equation for \( R \):**
- We can express \( R \) in terms of the angles:
\[ R \cos B \cos C = \frac{3}{\sqrt{5}} \]

7. **Relate \( R \) to \( \sin B \sin C \):**
- We can express \( R \sin B \sin C \) using the sine rule:
\[ R \sin B \sin C = \frac{3}{2\sqrt{5}} \]

8. **Calculate \( \sin B \sin C \):**
- From the previous equations, we can substitute \( R \):
\[ R = \frac{\sqrt{61}}{5} \]
- Thus, substituting \( R \) into the equation for \( \sin B \sin C \):
\[ \sin B \sin C = \frac{3}{2\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{61}} = \frac{9}{2\sqrt{61}} \]

9. **Final Result:**
- Therefore, the value of \( \sin B \sin C \) is:
\[ \sin B \sin C = \frac{9}{2\sqrt{61}} \]