Answer:
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