The radical centre of three circles described on the three sides `4x-7y + 10 = 0`, `x + y = 5` and `7x + 4y = 15` of a triangle as diameters is

Options

A. (a)(2, 3)
B. (b)(2, 1)
C. (c)(3, 2)
D. (d)(1, 2)

Answer

To find the radical center of the three circles described on the sides of the triangle formed by the lines \(4x - 7y + 10 = 0\), \(x + y = 5\), and \(7x + 4y = 15\), we can follow these steps:

### Step 1: Identify the equations of the lines
The equations of the lines are:
1. \(L_1: 4x - 7y + 10 = 0\)
2. \(L_2: x + y - 5 = 0\)
3. \(L_3: 7x + 4y - 15 = 0\)

### Step 2: Find the slopes of the lines
The slope \(m\) of a line given by the equation \(Ax + By + C = 0\) is given by \(-\frac{A}{B}\).

- For \(L_1\):
\[ m_1 = -\frac{4}{-7} = \frac{4}{7} \]

- For \(L_2\):
\[ m_2 = -\frac{1}{1} = -1 \]

- For \(L_3\):
\[ m_3 = -\frac{7}{4} \]

### Step 3: Check for perpendicularity
To find the orthocenter, we need to check if any two lines are perpendicular. Two lines are perpendicular if the product of their slopes equals \(-1\).

- Checking \(L_1\) and \(L_2\):
\[ m_1 \cdot m_2 = \frac{4}{7} \cdot (-1) = -\frac{4}{7} \quad (\text{not perpendicular}) \]

- Checking \(L_1\) and \(L_3\):
\[ m_1 \cdot m_3 = \frac{4}{7} \cdot \left(-\frac{7}{4}\right) = -1 \quad (\text{perpendicular}) \]

Thus, \(L_1\) and \(L_3\) are perpendicular.

### Step 4: Find the intersection of \(L_1\) and \(L_3\)
To find the orthocenter, we need to solve the equations of \(L_1\) and \(L_3\) simultaneously.

1. From \(L_1\):
\[ 4x - 7y + 10 = 0 \implies 4x - 7y = -10 \quad (1) \]

2. From \(L_3\):
\[ 7x + 4y - 15 = 0 \implies 7x + 4y = 15 \quad (2) \]

### Step 5: Solve the equations
We can multiply equation (1) by 4 and equation (2) by 7 to eliminate \(y\):

- Multiply (1) by 4:
\[ 16x - 28y = -40 \quad (3) \]

- Multiply (2) by 7:
\[ 49x + 28y = 105 \quad (4) \]

Now, add equations (3) and (4):
\[ 16x - 28y + 49x + 28y = -40 + 105 \]
\[ 65x = 65 \implies x = 1 \]

### Step 6: Substitute \(x\) back to find \(y\)
Substituting \(x = 1\) into equation (1):
\[ 4(1) - 7y + 10 = 0 \implies 4 - 7y + 10 = 0 \implies 14 - 7y = 0 \implies 7y = 14 \implies y = 2 \]

### Conclusion
The coordinates of the radical center (orthocenter) are:
\[ (x, y) = (1, 2) \]

### Final Answer
The radical center of the three circles is \((1, 2)\).