A point P(x,y) moves that the sum of its distance from the lines 2x-y-3=0 and x+3y+4=0 is 7. The area bounded by locus P is (in sq. unit)

Options

A. 70
B. `70sqrt(2)`
C. `35sqrt(2)`
D. 140

Explanation

To solve the problem, we need to find the area bounded by the locus of points \( P(x, y) \) such that the sum of its distances from the two given lines is equal to 7.

### Step 1: Identify the lines and their equations
The given lines are:
1. \( 2x - y - 3 = 0 \)
2. \( x + 3y + 4 = 0 \)

### Step 2: Write the distance formula from a point to a line
The distance \( d \) from a point \( P(x_1, y_1) \) to a line \( ax + by + c = 0 \) is given by:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]

### Step 3: Calculate the distances from point \( P(x, y) \) to each line
For the first line \( 2x - y - 3 = 0 \):
- Here, \( a = 2, b = -1, c = -3 \)
- The distance \( d_1 \) from point \( P(x, y) \) is:
\[ d_1 = \frac{|2x - y - 3|}{\sqrt{2^2 + (-1)^2}} = \frac{|2x - y - 3|}{\sqrt{5}} \]

For the second line \( x + 3y + 4 = 0 \):
- Here, \( a = 1, b = 3, c = 4 \)
- The distance \( d_2 \) from point \( P(x, y) \) is:
\[ d_2 = \frac{|x + 3y + 4|}{\sqrt{1^2 + 3^2}} = \frac{|x + 3y + 4|}{\sqrt{10}} \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the sum of the distances from point \( P(x, y) \) to the two lines is equal to 7:
\[ d_1 + d_2 = 7 \]
Substituting the distances:
\[ \frac{|2x - y - 3|}{\sqrt{5}} + \frac{|x + 3y + 4|}{\sqrt{10}} = 7 \]

### Step 5: Clear the denominators
Multiply through by \( \sqrt{10} \) to eliminate the denominators:
\[ \sqrt{10} \cdot \frac{|2x - y - 3|}{\sqrt{5}} + |x + 3y + 4| = 7\sqrt{10} \]
This simplifies to:
\[ \frac{\sqrt{10}}{\sqrt{5}} |2x - y - 3| + |x + 3y + 4| = 7\sqrt{10} \]
Since \( \frac{\sqrt{10}}{\sqrt{5}} = \sqrt{2} \):
\[ \sqrt{2} |2x - y - 3| + |x + 3y + 4| = 7\sqrt{10} \]

### Step 6: Analyze the absolute values
We need to consider different cases based on the signs of the expressions inside the absolute values. This will yield multiple linear equations.

1. **Case 1**: Both expressions are positive.
2. **Case 2**: First is positive, second is negative.
3. **Case 3**: First is negative, second is positive.
4. **Case 4**: Both expressions are negative.

### Step 7: Find the equations for each case
For example, in Case 1, we have:
\[ \sqrt{2} (2x - y - 3) + (x + 3y + 4) = 7\sqrt{10} \]
This will yield a linear equation in \( x \) and \( y \).

### Step 8: Find the area bounded by the locus
The equations derived from the cases will represent lines in the coordinate plane. The area bounded by these lines can be calculated using the formula for the area of a parallelogram or polygon formed by the intersection of these lines.

### Step 9: Calculate the area
Using the area formula for a parallelogram formed by two pairs of parallel lines:
\[ \text{Area} = \frac{|c_1 - c_2| \times |d_1 - d_2|}{|a_1b_2 - a_2b_1|} \]
Substituting the appropriate values will yield the area.

### Final Answer
The area bounded by the locus \( P \) is \( 70\sqrt{2} \) square units.