Question

Write the equation in standard form, give the center, and give the radius
x^2+y^2+4x-8y=5

Answer 1

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).