Translation of Axes - Hyperbola
Standard Equation for Hyperbolas:
where b2 = c2 - a2
vertices (h ± a,0) (0, k ± a)foci (h ± c,0) (0, k ± c)</span>
a is always larger than b; and a,b, and c are related by c2 = a2 + b2
ex.
Graph 9x<sup>2</sup> - 25y<sup>2 </sup>-54x + 250y -769 = 09x<sup>2 </sup>- 54x - 25y<sup>2 </sup>+ 250y = 769(9x<sup>2</sup> - 54x ) - ( 25y<sup>2 </sup>- 250y ) = 7699(x<sup>2 </sup>- 6x + 9) - 25(y<sup>2 </sup>- 10y + 25) = 769 +81 - 6259(x - 3)<sup>2</sup> - 25(y -5)<sup>2 </sup>= 225</span>
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)