Answer:
Step-by-step explanation:
x^2 + y^2 + 8x - 8y + 24 = 0 should be rewritten in preparation for "completing the square:" x^2 + 8x + y^2 - 8y + 24 = 0.
To complete the square of x^2 + 8x, take half of the coefficient (8) of the x term. Square this result (square 4), obtaining 16. Add 16 and then immediately subtract 16: you'll get x^2 + 8x + 16 - 16.
Doing the same thing with y^2 - 8y, you'll get:
y^2 - 8y + 16 - 16.
Putting this all back together: from x^2 + y^2 + 8x - 8y + 24 = 0
you'll get x^2 + 8x + 16 - 16 + y^2 - 8y + 16 - 16 + 24 = 0.
Note that the trinomial squares x^2 + 8x + 16 and y^2 - 8y + 16 can be rewritten as the squares of binomials:
(x + 4)^2 - 16 + (y - 4)^2 - 16 + 24 = 0.
Gathering the constant terms together on the right, we obtain:
(x + 4)^2 + (y - 4)^2 = 8.
comparing this result to (x - h)^2 + (y - k)^2 = r^2,
we see that h = -4, k = 4 and r^2 = 8.
Statement A is incorrect; the radius is 2√2.
Statement B is correct; see A (above).
Statement C is incorrect (see "h = -4, k = 4" above)
Statement D is correct.
