If area of the parallelogram formed by the lines `x+3y-a=0, 3x-2y+3a=0, x+3y+4a=0` and `3x-2y+7a=0` is 220 sq. units, then value of a is ________

Answer

To find the value of \( a \) such that the area of the parallelogram formed by the lines \( x + 3y - a = 0 \), \( 3x - 2y + 3a = 0 \), \( x + 3y + 4a = 0 \), and \( 3x - 2y + 7a = 0 \) is 220 square units, we can follow these steps:

### Step 1: Identify the pairs of parallel lines
The lines \( x + 3y - a = 0 \) and \( x + 3y + 4a = 0 \) are parallel because they have the same coefficients for \( x \) and \( y \). Similarly, the lines \( 3x - 2y + 3a = 0 \) and \( 3x - 2y + 7a = 0 \) are also parallel.

### Step 2: Calculate the distance between the parallel lines
The distance \( d \) between two parallel lines of the form \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
For the first pair of lines:
- \( C_1 = -a \)
- \( C_2 = 4a \)

Thus, the distance \( d_1 \) between these lines is:
\[ d_1 = \frac{|4a - (-a)|}{\sqrt{1^2 + 3^2}} = \frac{|4a + a|}{\sqrt{10}} = \frac{5a}{\sqrt{10}} = \frac{5a}{\sqrt{10}} = \frac{5a\sqrt{10}}{10} = \frac{a\sqrt{10}}{2} \]

For the second pair of lines:
- \( C_1 = 3a \)
- \( C_2 = 7a \)

Thus, the distance \( d_2 \) between these lines is:
\[ d_2 = \frac{|7a - 3a|}{\sqrt{3^2 + (-2)^2}} = \frac{|4a|}{\sqrt{9 + 4}} = \frac{4a}{\sqrt{13}} \]

### Step 3: Calculate the area of the parallelogram
The area \( A \) of the parallelogram formed by these lines is given by:
\[ A = d_1 \times d_2 \]
Substituting the distances we calculated:
\[ A = \left(\frac{a\sqrt{10}}{2}\right) \times \left(\frac{4a}{\sqrt{13}}\right) = \frac{4a^2\sqrt{10}}{2\sqrt{13}} = \frac{2a^2\sqrt{10}}{\sqrt{13}} \]

### Step 4: Set the area equal to 220 and solve for \( a \)
We know that the area is given as 220 square units:
\[ \frac{2a^2\sqrt{10}}{\sqrt{13}} = 220 \]
Multiplying both sides by \( \sqrt{13} \):
\[ 2a^2\sqrt{10} = 220\sqrt{13} \]
Dividing both sides by 2:
\[ a^2\sqrt{10} = 110\sqrt{13} \]
Now, squaring both sides:
\[ a^4 \cdot 10 = 12100 \cdot 13 \]
Calculating \( 12100 \cdot 13 \):
\[ 12100 \cdot 13 = 157300 \]
Thus:
\[ 10a^4 = 157300 \]
Dividing by 10:
\[ a^4 = 15730 \]
Taking the fourth root:
\[ a = \pm \sqrt[4]{15730} \]

### Step 5: Final value of \( a \)
Calculating \( \sqrt[4]{15730} \):
\[ a \approx \pm 11.11 \]

Thus, the possible values of \( a \) are approximately \( 11.11 \) and \( -11.11 \).