Final answer:
No, the equation of the circle is not in standard form. To rewrite it in standard form, we need to complete the square for both the x and y terms.
Step-by-step explanation:
No, the equation of the circle is not in standard form. The standard form of the equation of a circle is given by (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r represents the radius. To rewrite the given equation in standard form, we need to complete the square for both the x and y terms. Let's go through the steps:
1. Move the constant term to the right side of the equation:
x^2 + y^2 - 23x - 3y = -2
2. Group the x terms and complete the square:
(x^2 - 23x) + (y^2 - 3y) = -2
To complete the square for the x terms, we take half of the coefficient of x, square it, and add it inside the parentheses:
(x^2 - 23x + 529) + (y^2 - 3y) = -2 + 529
3. Group the y terms and complete the square:
(x^2 - 23x + 529) + (y^2 - 3y + 9/4) = -2 + 529 + 9/4
To complete the square for the y terms, we take half of the coefficient of y, square it, and add it inside the parentheses:
(x^2 - 23x + 529) + (y^2 - 3y + 9/4) = -2 + 529 + 9/4 + 36/4
4. Simplify and rewrite the equation in standard form:
(x - 23/2)^2 + (y - 3/2)^2 = 529/4 + 9/4 + 36/4 - 2
(x - 23/2)^2 + (y - 3/2)^2 = 572/4
(x - 23/2)^2 + (y - 3/2)^2 = 143