The coordinates of points A,B,C and D are `(-3, 5), (4, -2), (x, 3x)` and (6, 3) respectively and Area of `(Delta ABC)/(Delta BCD)=(2)/(3)` ,find x.

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 ratio of the areas of triangles \( ABC \) and \( BCD \).

### Step 1: Identify the coordinates
The coordinates of the points are:
- \( A(-3, 5) \)
- \( B(4, -2) \)
- \( C(x, 3x) \)
- \( D(6, 3) \)

### Step 2: Area of triangle \( ABC \)
The area of triangle \( ABC \) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of points \( A, B, \) and \( C \):
\[ \text{Area}_{ABC} = \frac{1}{2} \left| -3(-2 - 3x) + 4(3x - 5) + x(5 - (-2)) \right| \]
Calculating this step by step:
\[ = \frac{1}{2} \left| -3(-2 - 3x) + 4(3x - 5) + x(5 + 2) \right| \]
\[ = \frac{1}{2} \left| 6 + 9x + 12x - 20 + 7x \right| \]
\[ = \frac{1}{2} \left| 28x - 14 \right| \]
\[ = 14|x - \frac{1}{2}| \]

### Step 3: Area of triangle \( BCD \)
Now, we calculate the area of triangle \( BCD \) using the same formula:
\[ \text{Area}_{BCD} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of points \( B, C, \) and \( D \):
\[ \text{Area}_{BCD} = \frac{1}{2} \left| 4(3x - 3) + x(3 - (-2)) + 6(-2 - 3x) \right| \]
Calculating this step by step:
\[ = \frac{1}{2} \left| 12x - 12 + 5x - 12 - 18x \right| \]
\[ = \frac{1}{2} \left| -x - 24 \right| \]
\[ = \frac{1}{2} |x + 24| \]

### Step 4: Set up the ratio of areas
We know from the problem statement that:
\[ \frac{\text{Area}_{ABC}}{\text{Area}_{BCD}} = \frac{2}{3} \]
Substituting the areas we calculated:
\[ \frac{14|x - \frac{1}{2}|}{\frac{1}{2}|x + 24|} = \frac{2}{3} \]
Cross-multiplying gives:
\[ 42|x - \frac{1}{2}| = |x + 24| \]

### Step 5: Solve the equation
We will consider two cases for the absolute values.

**Case 1:** \( 42(x - \frac{1}{2}) = x + 24 \)
\[ 42x - 21 = x + 24 \]
\[ 41x = 45 \implies x = \frac{45}{41} \]

**Case 2:** \( 42(x - \frac{1}{2}) = -(x + 24) \)
\[ 42x - 21 = -x - 24 \]
\[ 43x = 3 \implies x = -\frac{3}{43} \]

### Conclusion
The values of \( x \) that satisfy the given condition are:
\[ x = \frac{45}{41} \quad \text{and} \quad x = -\frac{3}{43} \]