(1) The equation of two diameters of the circle passing Brough the point \( (3,4) \) are \( x+y=14 \) and \( 2 x-y=4 \) Find the equation of the circle

To find the equation of the circle, follow these steps: 1. **Determine the Center of the Circle:** The two given diameters of the circle are: \[ x + y = 14 \quad \text{(Diameter 1)} \] \[ 2x - y = 4 \quad \text{(Diameter 2)} \] The center of the circle is the intersection point of these two diameters. Solve the system of equations to find the center \((h, k)\): \[ \begin{cases} x + y = 14 \\ 2x - y = 4 \end{cases} \] **Adding the two equations:** \[ (x + y) + (2x - y) = 14 + 4 \\ 3x = 18 \\ x = 6 \] **Substitute \(x = 6\) into the first equation:** \[ 6 + y = 14 \\ y = 8 \] So, the center of the circle is at \((6, 8)\). 2. **Find the Radius of the Circle:** The circle passes through the point \((3, 4)\). Use the distance formula to find the radius \(r\): \[ r = \sqrt{(3 - 6)^2 + (4 - 8)^2} \\ r = \sqrt{(-3)^2 + (-4)^2} \\ r = \sqrt{9 + 16} \\ r = \sqrt{25} \\ r = 5 \] 3. **Write the Equation of the Circle:** The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substituting the values we found: \[ (x - 6)^2 + (y - 8)^2 = 5^2 \\ (x - 6)^2 + (y - 8)^2 = 25 \] **Alternatively**, expanding the equation: \[ x^2 - 12x + 36 + y^2 - 16y + 64 = 25 \\ x^2 + y^2 - 12x - 16y + 100 = 25 \\ x^2 + y^2 - 12x - 16y + 75 = 0 \] **Final Answer:** The equation of the circle is \[ (x - 6)^2 + (y - 8)^2 = 25. \]