The co-ordinates of A,B,C are (6,3) ,(-3,5) , (4,-2) respectively and P is a point (x,y) , then the value of `(DeltaPBC)/(DeltaABC)`:

Options

A. a. `(x+y-2)/(7)`
B. b. `(x-y)/(4)`
C. c. `(x+y-3)/(5)`
D. d. none of these

Answer

To find the value of \(\frac{\Delta PBC}{\Delta ABC}\), we need to calculate the areas of triangles \(PBC\) and \(ABC\) using the coordinates provided.

### Step 1: Calculate the Area of Triangle ABC

The coordinates of points \(A\), \(B\), and \(C\) are:
- \(A(6, 3)\)
- \(B(-3, 5)\)
- \(C(4, -2)\)

The formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:

\[ \Delta = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Substituting the coordinates of points \(A\), \(B\), and \(C\):

\[ \Delta ABC = \frac{1}{2} \left| 6(5 - (-2)) + (-3)(-2 - 3) + 4(3 - 5) \right| \]

Calculating each term:

1. \(6(5 + 2) = 6 \times 7 = 42\)
2. \(-3(-2 - 3) = -3 \times (-5) = 15\)
3. \(4(3 - 5) = 4 \times (-2) = -8\)

Now, substituting these values back into the area formula:

\[ \Delta ABC = \frac{1}{2} \left| 42 + 15 - 8 \right| = \frac{1}{2} \left| 49 \right| = \frac{49}{2} \]

### Step 2: Calculate the Area of Triangle PBC

Let \(P\) have coordinates \((x, y)\). The area of triangle \(PBC\) is given by:

\[ \Delta PBC = \frac{1}{2} \left| x(5 - (-2)) + (-3)(-2 - y) + 4(y - 5) \right| \]

Substituting the values:

\[ \Delta PBC = \frac{1}{2} \left| x(7) + (-3)(-2 - y) + 4(y - 5) \right| \]

Calculating each term:

1. \(x(7) = 7x\)
2. \(-3(-2 - y) = 6 + 3y\)
3. \(4(y - 5) = 4y - 20\)

Now, substituting these values back into the area formula:

\[ \Delta PBC = \frac{1}{2} \left| 7x + 6 + 3y + 4y - 20 \right| = \frac{1}{2} \left| 7x + 7y - 14 \right| \]

### Step 3: Calculate the Ratio \(\frac{\Delta PBC}{\Delta ABC}\)

Now we can find the ratio:

\[ \frac{\Delta PBC}{\Delta ABC} = \frac{\frac{1}{2} \left| 7x + 7y - 14 \right|}{\frac{49}{2}} = \frac{\left| 7x + 7y - 14 \right|}{49} \]

We can simplify this further:

\[ \frac{\Delta PBC}{\Delta ABC} = \frac{7(x + y - 2)}{49} = \frac{x + y - 2}{7} \]

### Final Answer

Thus, the value of \(\frac{\Delta PBC}{\Delta ABC}\) is:

\[ \frac{x + y - 2}{7} \]