The equation of a circle is given in general form. Complete parts a through d to follow the step by step process to sketch the graph of the circle. \( x^{2}+y^{2}-14 x+2 y+34=0 \) a) Write the equation of the circle in standard form. The equation is \( (x-7)^{2}+(y+1)^{2}=16 \). (Simplify your answer. \( ) \) b) Determine the center and radius of the circle. The center is

To find the center and radius of the circle from the standard form of the equation, we can analyze the equation you provided: ### a) Write the equation of the circle in standard form. You have already completed this part, and the equation in standard form is: \[ (x - 7)^2 + (y + 1)^2 = 16 \] ### b) Determine the center and radius of the circle. From the standard form of the equation of a circle, which is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] where \((h, k)\) is the center of the circle and \(r\) is the radius, we can identify the following: - The center \((h, k)\) is \((7, -1)\). - The radius \(r\) is the square root of \(16\), which is \(4\). Thus, the center is: \[ \text{Center: } (7, -1) \] And the radius is: \[ \text{Radius: } 4 \] ### c) Sketch the graph of the circle. To sketch the graph of the circle, follow these steps: 1. **Plot the center**: Mark the point \((7, -1)\) on the coordinate plane. 2. **Draw the radius**: From the center, measure \(4\) units in all directions (up, down, left, right) to find the points on the circle. - Up: \((7, -1 + 4) = (7, 3)\) - Down: \((7, -1 - 4) = (7, -5)\) - Left: \((7 - 4, -1) = (3, -1)\) - Right: \((7 + 4, -1) = (11, -1)\) 3. **Draw the circle**: Connect these points with a smooth curve to form the circle. ### d) Identify any additional properties (if needed). If you need to identify any additional properties, such as the intercepts or the area, please let me know!