A circle rolls between pair of lines `9x^(2)+24xy +16y^(2) - 25 =0` touching both of them. Then its area is

Options

A. `4pi` sq. units
B. `8pi` sq. units
C. `12pi` sq. units
D. `pi` sq. units

Explanation

To solve the problem step by step, we need to find the area of the circle that rolls between the given pair of lines.

### Step 1: Identify the equation of the pair of lines
The given equation is:
\[ 9x^2 + 24xy + 16y^2 - 25 = 0 \]

### Step 2: Rewrite the equation in a factored form
We can rewrite the equation as:
\[ 9x^2 + 12xy + 12xy + 16y^2 = 25 \]
This can be factored as:
\[ (3x + 4y)^2 = 25 \]

### Step 3: Solve for the lines
Taking the square root of both sides gives us:
\[ 3x + 4y = 5 \quad \text{and} \quad 3x + 4y = -5 \]
Thus, the two lines are:
1. \( 3x + 4y = 5 \)
2. \( 3x + 4y = -5 \)

### Step 4: Determine 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 our lines:
- \( C_1 = -5 \)
- \( C_2 = 5 \)
- \( A = 3 \)
- \( B = 4 \)

Substituting these values into the formula gives:
\[ d = \frac{|5 - (-5)|}{\sqrt{3^2 + 4^2}} = \frac{10}{\sqrt{9 + 16}} = \frac{10}{\sqrt{25}} = \frac{10}{5} = 2 \]

### Step 5: Relate the distance to the diameter of the circle
Since the distance between the lines is equal to the diameter of the circle, we have:
\[ 2r = 2 \]
Thus, the radius \( r \) of the circle is:
\[ r = 1 \]

### Step 6: Calculate the area of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi (1^2) = \pi \]

### Final Answer
The area of the circle is:
\[ \pi \, \text{square units} \]