Final answer:
To write the equation in standard form, complete the square for both the x and y variables. The center of the circle is (-2, 4), and the radius is sqrt(17).
Step-by-step explanation:
To write the equation in standard form, we need to complete the square for both the x and y variables. Starting with the given equation:
x^2 + y^2 + 4x - 8y = 5
Let's group the x terms and y terms:
(x^2 + 4x) + (y^2 - 8y) = 5
To complete the square for the x terms, we take half of the coefficient of x, square it, and add it to both sides:
(x^2 + 4x + (4/2)^2) + (y^2 - 8y) = 5 + (4/2)^2
(x^2 + 4x + 4) + (y^2 - 8y) = 9
Similarly, for the y terms:
(x^2 + 4x + 4) + (y^2 - 8y + (-8/2)^2) = 9 + (-8/2)^2
(x^2 + 4x + 4) + (y^2 - 8y + 16) = 17
Now, we can rewrite the equation in standard form:
(x + 2)^2 + (y - 4)^2 = 17
The equation is now in standard form. The center of the circle is (-2, 4), and the radius is sqrt(17).