If the co-ordinates of points A,B,C,D are (6,3),(-3,5),(4,-2) and (x,3x) respectively and if `(Delta DBC)/(DeltaABC)=1/2`, then x=

Options

A. `8/11`
B. `11/8`
C. `7/9`
D. `0`

Answer

To solve the problem, we need to find the value of \( x \) given the coordinates of points A, B, C, and D, and the condition that the ratio of the areas of triangles DBC and ABC is \( \frac{1}{2} \).

### Step-by-Step Solution:

1. **Identify the Coordinates:**
- Point A: \( (6, 3) \)
- Point B: \( (-3, 5) \)
- Point C: \( (4, -2) \)
- Point D: \( (x, 3x) \)

2. **Area of Triangle Formula:**
The area of a triangle formed by points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

3. **Calculate Area of Triangle DBC:**
Using the coordinates of points D, B, and C:
\[ \text{Area}_{DBC} = \frac{1}{2} \left| x(5 - (-2)) + (-3)(-2 - 3x) + 4(3x - 5) \right| \]
Simplifying:
\[ = \frac{1}{2} \left| x(7) + 3(-2 - 3x) + 4(3x - 5) \right| \]
\[ = \frac{1}{2} \left| 7x - 6 - 9x + 12x - 20 \right| \]
\[ = \frac{1}{2} \left| 10x - 26 \right| \]

4. **Calculate Area of Triangle ABC:**
Using the coordinates of points A, B, and C:
\[ \text{Area}_{ABC} = \frac{1}{2} \left| 6(5 - (-2)) + (-3)(-2 - 3) + 4(3 - 5) \right| \]
Simplifying:
\[ = \frac{1}{2} \left| 6(7) + 3(1) + 4(-2) \right| \]
\[ = \frac{1}{2} \left| 42 + 3 - 8 \right| \]
\[ = \frac{1}{2} \left| 37 \right| = \frac{37}{2} \]

5. **Set Up the Ratio of Areas:**
Given that \( \frac{\text{Area}_{DBC}}{\text{Area}_{ABC}} = \frac{1}{2} \):
\[ \frac{\frac{1}{2} |10x - 26|}{\frac{37}{2}} = \frac{1}{2} \]
Simplifying:
\[ |10x - 26| = \frac{37}{2} \]

6. **Solve for \( x \):**
This gives us two equations:
\[ 10x - 26 = \frac{37}{2} \quad \text{and} \quad 10x - 26 = -\frac{37}{2} \]

**First Equation:**
\[ 10x = \frac{37}{2} + 26 = \frac{37}{2} + \frac{52}{2} = \frac{89}{2} \]
\[ x = \frac{89}{20} = \frac{11}{8} \]

**Second Equation:**
\[ 10x = -\frac{37}{2} + 26 = -\frac{37}{2} + \frac{52}{2} = \frac{15}{2} \]
\[ x = \frac{15}{20} = \frac{3}{4} \]

7. **Final Values of \( x \):**
The two possible values for \( x \) are \( \frac{11}{8} \) and \( \frac{3}{8} \).

### Conclusion:
Thus, the values of \( x \) are \( \frac{11}{8} \) and \( \frac{3}{8} \).