Standard Equation for Hyperbolas:

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

vertices (h Â± a,0) (0, k Â± a)foci (h Â± c,0) (0, k Â± c)</span>

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

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 Â± 5,5)

ex.

16x^{2} - 9y^{2}- 224x - 54y + 847 = 0

16x^{2} - 224x -9y^{2} - 54y = -847

(16x^{2} - 224x ) - (9y^{2} - 54y ) = -847

16( x^{2} - 14x + 49) - 9( y^{2} + 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 Â± 4)

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

c^{2} = 16 + 9

c^{2} = 25

c = 5

foci (7,-3 Â± 5)