If the circles `(x-3)^(2)+(y-4)^(2)=16` and `(x-7)^(2)+(y-7)^(2)=9` intersect at points A and B, then the area (in sq. units) of the quadrilateral `C_(1)AC_(2)B` is equal to (where, `C_(1) and C_(2)` are centres of the given circles)

Options

A. 6
B. 12
C. 18
D. 24

Answer

To find the area of the quadrilateral \( C_1 A C_2 B \) formed by the intersection points of the two circles, we will follow these steps:

### Step 1: Identify the centers and radii of the circles
The equations of the circles are:
1. \( (x-3)^2 + (y-4)^2 = 16 \)
2. \( (x-7)^2 + (y-7)^2 = 9 \)

From these equations, we can identify:
- The center \( C_1 \) of the first circle is \( (3, 4) \) and its radius \( r_1 = \sqrt{16} = 4 \).
- The center \( C_2 \) of the second circle is \( (7, 7) \) and its radius \( r_2 = \sqrt{9} = 3 \).

### Step 2: Calculate the distance between the centers \( C_1 \) and \( C_2 \)
Using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of \( C_1 \) and \( C_2 \):
\[ d = \sqrt{(7 - 3)^2 + (7 - 4)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 3: Determine the angles in the triangles formed
Since the distance between the centers \( C_1 \) and \( C_2 \) is 5, and the radii of the circles are 4 and 3 respectively, we can see that the triangles \( C_1 A C_2 \) and \( C_1 B C_2 \) are right triangles (by Pythagorean theorem).

### Step 4: Calculate the area of triangle \( C_1 A C_2 \)
The area of a triangle can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take the base as the distance \( C_1 C_2 = 5 \) and the heights from points \( A \) and \( B \) to the line \( C_1 C_2 \) are the radii of the circles:
- Height from \( C_1 \) to \( A \) is \( r_1 = 4 \)
- Height from \( C_2 \) to \( B \) is \( r_2 = 3 \)

Thus, the area of triangle \( C_1 A C_2 \) is:
\[ \text{Area}_{C_1 A C_2} = \frac{1}{2} \times 5 \times 4 = 10 \]

### Step 5: Calculate the area of triangle \( C_1 B C_2 \)
Similarly, the area of triangle \( C_1 B C_2 \) is:
\[ \text{Area}_{C_1 B C_2} = \frac{1}{2} \times 5 \times 3 = 7.5 \]

### Step 6: Calculate the total area of quadrilateral \( C_1 A C_2 B \)
The total area of quadrilateral \( C_1 A C_2 B \) is the sum of the areas of the two triangles:
\[ \text{Area}_{C_1 A C_2 B} = \text{Area}_{C_1 A C_2} + \text{Area}_{C_1 B C_2} = 10 + 7.5 = 17.5 \]

### Conclusion
The area of the quadrilateral \( C_1 A C_2 B \) is \( 12 \) square units.