I am lost on this one!!!

Question

Determine values for A, B, and C such that the equation below represents the
given type of conic. Each axis of the ellipse, parabola, and hyperbola should
be horizontal or vertical. Then rewrite your equation for each conic in standard
form, identify (h, k), and describe the translation. Part A: Circle, Part B: Ellipse,
Part C: Parabola, Part D: Hyperbola
Ax^2+Bxy+Cy^2+2x-4y-5=0

Answer 1

Circle: A = 1 , C = 1 , B = 0
x^2 + 2x + y^2 - 4y = 5
x^2 + 2x + 1 + y^2 - 4y + 4 = 5+1+4
(x+1)^2 + (y-2)^2 = 10
circle: centre(-1,2) and radius √10

Ellipse : A=1, C=2, B=0
x^2 + 2x + 2(y^2 - 2y ) = 5
x^2 + 2x + 1 + 2(y^2 - 2y + 1) = 5 + 1 + 2
(x+1)^2 + 2(y-1)^2 = 8
divide by 8
(x+1)^2 /8 + (y-1)^2 /4 = 1

Parabola:
let A=1, C=0, B=0
x^2 + 2x - 5 = 4y
4y = x^2 + 2x + 1 - 1 - 5
4y = (x+1)^2 - 6
y = (1/4)(x+1)^2 - 3/2

hyperbola:
let A=1, B=0, C=-1
x^2 + 2x - (y^2 + 2y) = 5
(x^2 + 2x + 1) - (y^2 + 2y + 1) = 5 + 1 - 1
(x+1)^2 - (y+1)^2 = 5

(x+1)^2 /5 - (y+1)^2 /5 = 1

Since you wanted all axes to be either vertical or horizontal, the xy term cannot be there, so in all cases B = 0

Answer 2

Thank you Reiny!!!! I've been asking for this as well......