Conic Sections Translation of Axes:

Ellipse:

standard equations for ellipse:

where b² = a² - c²

center: (h,k) (h,k)major axis: x = h, length 2a y = k, length 2aminor axis: y = k, length 2b x = h, length 2bfoci: (h ± c,k) (h, k ± c)vertices: (h ± a,k) (h, k ± a)covertices: (h, k ± b) (h ± b,k)

a is always larger than b; and a,b, and c are related by c² = a² - b²

ex.
graph 16x² + 25y² - 64x-200y + 64 = 0
convert to standard form
16x² - 64x + 25y² -200y = - 64
complete the square for the x and y terms

(16x² - 64x ) + (25y² -200y ) = - 64
16(x² - 4x ) + 25(y² - 8y ) = -64 complete the square
16(x²- 4x + 4) + 25(y²- 8y + 16) = -64 + 64 + 400
16(x - 2)² + 25(y - 4)² = 400

a = 5; b = 4
center: (2,4)
major axis: x = 2, length 10
minor axis: y = 4, length 8
c² = a² - b²
c² = 25 - 16
c² = 9
c = 3

foci: (2 ± 3,4) (5,4) and (-1,4)vertices: (2 ± 5,4) (7,4) and (-3,4)covertices: (2, 4 ± 4) (2,8) and (2,0)

ex.
graph 25x² + 9y²+ 200x + 54y + 256 = 0
convert to standard form
25x² +200x + 9y²+ 4y = -256
complete the square for the x and y terms

(25x²+200x )+ (9y²+ 54y ) = - 256
25(x² + 8x ) + 9(y² + 6y ) = -256
25(x² + 8x + 16) + 9(y² + 6y + 9) = -256 + 400 + 81
25(x + 4)² + 9(y + 3)² = 225

a = 5; b = 3
center: (-4,-3)
major axis: y = -3, length 10
minor axis: x = -4, length 6
c² = a² - b²
c² = 25 - 9
c² = 16
c = 4

foci: (-4, -3 ± 4) (-4,1) and (-4,-7)vertices: (-4, -3 ± 5) (-4,2) and (-4,-8)covertices: (-4 ± 3,-3) (-1,-3) and (-7,-3)