Conic Sections: Translation of Axes:
The general equation for conic sections is :
Ax2 + By2 + Dx + Ey + F = 0
were A and B are not zero.
If AB = 0 then it is a parabola
If AB > 0 then it is an ellipse
If AB < 0 then it is a hyperbola
Conic Sections:
Hyperbola:
The standard equation for hyperbolas is:
where b2 = c2 - a2
vertices (± a,0) (0, ± a)
foci (± c,0) (0, ± c)
transverse
axis on x-axis, on y-axis,
length 2a length 2a
conjugate
axis on y-axis, on x-axis,
length 2b length 2b
a is always larger than b; and a,b, and c are related by c2 = a2 + b2
ex.
Graph 9x2 - 16y2= 144
a2 = 16 ; b2 = 9
major axis: x-axis
vertices: (± 4,0)
c2 = a+ b
c2 = 16 + 9
c2 = 25
foci: (± 5,0)
Graph 36x2 - 4y2 + 144 = 0
36x2 - 4y2 = -144 factor -1 out
4y2 - 36x2 = 144
a2 = 36 ; b2 = 4
major axis: y-axis
vertices: (0,± 6)
c2= a2 + b2
c2= 36 + 4
c2 = 40
(to find the asymptotes, let the x term equal the y term and solve for y)
Standard Equation for Hyperbolas:
where b2 = c2 - a2
vertices (h ± a,0) (0, k ± a)
foci (h ± c,0) (0, k ± c)
a is always larger than b; and a,b, and c are related by c2 = a2 + b2
ex.
Graph 9x2 - 25y2 -54x + 250y -769 = 0
9x2 - 54x - 25y2 + 250y = 769
(9x2 - 54x ) - ( 25y2 - 250y ) = 769
9(x2 - 6x + 9) - 25(y2 - 10y + 25) = 769 +81 - 625
9(x - 3)2 - 25(y -5)2 = 225
a = 5 ; b = 3
Center (3,5)
asymptotes
vertices (3 ± 5,5)
ex.
16x2 - 9y2- 224x - 54y + 847 = 0
16x2 - 224x -9y2 - 54y = -847
(16x2 - 224x ) - (9y2 - 54y ) = -847
16( x2 - 14x + 49) - 9( y2 + 6y + 9) = -847 +784 -81
16( x - 7)2 - 9(y + 3)2 = -144
9(y + 3)2 - 16( x - 7)2 = 144 Factor -1 out of both sides
a = 4; b = 3
Center (7,-3)
vertices (7,-3 ± 4)
c2 = a2 + b2
c2 = 16 + 9
c2 = 25
c = 5
foci (7,-3 ± 5)
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