**Conic Sections Translation of Axes:**

**Ellipse:**

standard equations for ellipse:

where b^{2} = a^{2} - c^{2}

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 Â± c,k) (h, k Â± c)vertices: (h Â± a,k) (h, k Â± a)covertices: (h, k Â± b) (h Â± b,k)

a is always larger than b; and a,b, and c are related by c^{2 }= a^{2} - b^{2}

ex.

graph 16x^{2 }+ 25y^{2 }- 64x-200y + 64 = 0

convert to standard form

16x^{2 }- 64x + 25y^{2 }-200y = - 64

complete the square for the x and y terms

(16x^{2 }- 64x ) + (25y^{2 }-200y ) = - 64

16(x^{2 }- 4x ) + 25(y^{2 }- 8y ) = -64 complete the square

16(x^{2}- 4x + 4) + 25(y^{2}- 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

c^{2 }= a^{2 }- b^{2}

c^{2} = 25 - 16

c^{2} = 9

c = 3

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

ex.

graph 25x^{2 }+ 9y^{2}+ 200x + 54y + 256 = 0

convert to standard form

25x^{2 }+200x + 9y^{2}+ 4y = -256

complete the square for the x and y terms

(25x^{2}+200x )+ (9y^{2}+ 54y ) = - 256

25(x^{2 }+ 8x ) + 9(y^{2 }+ 6y ) = -256

25(x^{2} + 8x + 16) + 9(y^{2 }+ 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

c^{2 }= a^{2 }- b^{2}

c^{2} = 25 - 9

c^{2} = 16

c = 4

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