Find the eccentricity of 8x^2 + 6y^2 - 32x + 24y + 8 = 0

1)

x 2 + y 2 - 10x - 8y + 1 = 0.

(x^2-10x+25-25) + (y^2-8y+16-16) + 1 = 0

(x-5) ^2 - 25 + (y-4) ^2 - 16 + 1 = 0

(x-5) ^2 + (y-4) ^2 = 40

center (5,4); radius = sqrt (40)

2)

8x^2 + 6y 2 - 32x + 24y + 8 = 0.

8 (x^2-4x+4-4) + 6 (y^2+4y+4-4) + 8=0

8 (x-2) ^2 - 32 + 6 (y+2) ^2 - 24 + 8 = 0

8 (x-2) ^2+6 (y+2) ^2 = 48

divide throughout by 48

(x-2) ^2 / 6 + (y+2) ^2 / 8 = 1

Ellipse with center (2,-2)

a=sqrt (6)

b=sqrt (8)

c^2 = 8-6 = 2

c = sqrt (2)

eccentricity = c/a = sqrt (2) / sqrt (6)

3)

y=x^2-12x+36-36

y = (x-6) ^2 - 36

Vertex is (6,-36)

(x-h) ^2=4p (y-k)

(x-6) ^2 = 4p (y+36)

4p=1

p=1/4

focus : (h, k+p) = (6, - 36+1/4) = (6, - 143/4)

4)

focus lies on a vertical line, so the major axis is parallel to the y-axis

(x-h) ^2/a^2 + (y-b) ^2/b^2 = 1

h=-2

k=0

(x+2) ^2/a^2+y^2/b^2 = 1

2b=20

b=10

b^2=100

(x+2) ^2/a^2 + y^2/100 = 1

e=c/a

c/a = 4/5

c = (4/5) a

c^2 = 16/25 a^2

c^2 = b^2-a^2

(16/25) a^2 = 100 - a^2

a^2 (16/25+1) = 100

41a^2/25 = 100

a^2=2500/41

a = sqrt (2500/41)

(x+2) ^2/a^2 + y^2/100 = 1

(x+2) ^2 / [2500/41] + y^2/100 = 1