Determine The Type Of Conics Each Equation Represents Using The Discriminant

1. 2×2-4xy+8y2+7=0

2. 2xy-x+y-3=0

3.x2-y2+4=0

Solution


>0 hyperbola, <0 ellipse, =0 parabola

unless it’s a degenerate form such as

2 intersecting lines, a point, 2 lines

1 b^2 -ac = 2^2-2(8) =-12 <0 ellipse

2 b^2 -ac = 1^2 = 1 >0, hyperbola

3 is a another hyperbola, b^2 -ac =0^2 -(1)(-1)= 1 >0

x^2 – y^2 = -4

y^2 -x^2 = 4

y^2/2^2 – x^2/2^2 = 1

in y^2/a^2 -x^2/b^2 = 1 form

c^2 = a^2 +b^2, c =sqr8= 2sqr2

2 branches of the hyperbola are above and below the x axis

asymptotes are y=x and y=-x

mid point between the 2 branches is the origin

vertices are (0,2) and (0,-2)

foci are (0,2sqr2) and (0,-2sqr2)