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b2 = c2 − a2 we get 2 → 2 . This shows that y → ± x as |x| grows large. Thus y = ± x are the
asymptotes to the graph as predicted and our choice of labels for the endpoints of the conjugate
axis is justiﬁed. In our equation of the hyperbola we can substitute a2 − c2 = −b2 which yields As x and y attain very large values, the quantity a2 − c2 x2 + a2 y 2 = a2 a2 − c2
−b2 x2 + a2 y 2 = −a2 b2
x2 y 2
The equation above is for a hyperbola whose center is the origin and which opens to the left and
right. If the hyperbola were centered at a point (h, k ), we would get the following.
Equation 7.6. The Standard Equation of a Horizontala Hyperbola For positive numbers
a and b, the equation of a horizontal hyperbola with center (h, k ) is
(x − h)2 (y − k )2
a That is, a hyperbola whose branches open to the left and right If the roles of x and y were interchanged, then the hyperbola’s branches would open upwards and
downwards and we would get a ‘vertical’ hyperbola.
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