Convert the equation 1/2x2 + 1/2y2 = 3x - 5y - 9 to standard form and find the circle's radius and center.

Alright Alice — let's do this. To find the radius and center of the circle, we need to convert the given equation into the standard form of a circle's equation, which is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center and r is the radius. Step 1: Rearrange the equation to group x terms and y terms, and move the constant to the right side. The given equation is: (1)/(2)x^2 + (1)/(2)y^2 = 3x - 5y - 9 Move the x and y terms to the left side: (1)/(2)x^2 - 3x + (1)/(2)y^2 + 5y = -9 Step 2: Eliminate the fractional coefficients for x^2 and y^2 by multiplying the entire equation by 2. 2 ( (1)/(2)x^2 - 3x + (1)/(2)y^2 + 5y ) = 2(-9) x^2 - 6x + y^2 + 10y = -18 Step 3: Complete the square for the x terms and the y terms. To complete the square for x^2 - 6x, take half of the coefficient of x (which is -6), square it ((-3)^2 = 9), and add it to both sides. To complete the square for y^2 + 10y, take half of the coefficient of y (which is 10), square it ((5)^2 = 25), and add it to both sides. (x^2 - 6x + 9) + (y^2 + 10y + 25) = -18 + 9 + 25 Step 4: Rewrite the squared terms and simplify the right side. (x - 3)^2 + (y + 5)^2 = -18 + 34 (x - 3)^2 + (y + 5)^2 = 16 Step 5: Identify the center and radius from the standard form. Comparing (x - 3)^2 + (y + 5)^2 = 16 with (x-h)^2 + (y-k)^2 = r^2: The center (h, k) is (3, -5). The radius squared r^2 is 16, so the radius r = sqrt(16) = 4. The coordinates of the center of the circle are (3, -5) and the radius is 4. I cannot draw the circle in the grid, but you would plot the center at (3, -5) and then draw a circle with a radius of 4 units around that center. Drop the next question.