A circle has the equation \( 3(x-3)^{2}+3 y^{2}=3 \). (a) Find the center ( \( h, k) \) and radius \( r \) of the circle. (b) Graph the circle. (c) Find the intercepts, if any, of the graph. (a) The center of the circle is (Type an ordered pair, using integers or decimals.)

To solve the problem, we will start with part (a) and find the center and radius of the circle given by the equation \( 3(x-3)^{2}+3 y^{2}=3 \). ### Step 1: Simplify the equation First, we can simplify the equation by dividing all terms by 3: \[ (x-3)^{2} + y^{2} = 1 \] ### Step 2: Identify the center and radius The standard form of a circle's equation is: \[ (x-h)^{2} + (y-k)^{2} = r^{2} \] From our simplified equation \( (x-3)^{2} + y^{2} = 1 \), we can identify: - \( h = 3 \) - \( k = 0 \) - \( r^{2} = 1 \) which gives \( r = 1 \) Thus, the center of the circle is \( (3, 0) \) and the radius is \( 1 \). ### Answer for part (a) The center of the circle is \( (3, 0) \) and the radius is \( 1 \). Now, let's proceed to part (b) and graph the circle. ### Step 3: Graph the circle The center of the circle is at \( (3, 0) \) and the radius is \( 1 \). This means the circle will extend 1 unit in all directions from the center. ### Step 4: Find the intercepts To find the intercepts, we will set \( y = 0 \) to find the x-intercepts and \( x = 0 \) to find the y-intercepts. #### Finding x-intercepts: Set \( y = 0 \): \[ (x-3)^{2} + 0^{2} = 1 \] This simplifies to: \[ (x-3)^{2} = 1 \] Taking the square root of both sides gives: \[ x - 3 = \pm 1 \] Thus, we have: \[ x = 4 \quad \text{and} \quad x = 2 \] So the x-intercepts are \( (4, 0) \) and \( (2, 0) \). #### Finding y-intercepts: Set \( x = 0 \): \[ (0-3)^{2} + y^{2} = 1 \] This simplifies to: \[ 9 + y^{2} = 1 \] Rearranging gives: \[ y^{2} = 1 - 9 = -8 \] Since \( y^{2} = -8 \) has no real solutions, there are no y-intercepts. ### Summary of results - **Center**: \( (3, 0) \) - **Radius**: \( 1 \) - **X-intercepts**: \( (4, 0) \) and \( (2, 0) \) - **Y-intercepts**: None Now, I will graph the circle and summarize the intercepts. ### Graphing the circle The graph will show a circle centered at \( (3, 0) \) with a radius of \( 1 \), touching the x-axis at \( (4, 0) \) and \( (2, 0) \). ### Final answers (a) The center of the circle is \( (3, 0) \). (b) The graph of the circle is centered at \( (3, 0) \) with radius \( 1 \). (c) The intercepts are \( (4, 0) \) and \( (2, 0) \) for x-intercepts, and there are no y-intercepts.