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Unformatted text preview: The Standard Equation of a Vertical Hyperbola For positive numbers a
and b, the equation of a vertical hyperbola with center (h, k ) is:
(y − k )2 (x − h)2
The values of a and b determine how far in the x and y directions, respectively, one counts from the
center to determine the rectangle through which the asymptotes pass. In both cases, the distance 7.5 Hyperbolas 437 √
from the center to the foci, c, as seen in the derivation, can be found by the formula c = a2 + b2 .
Lastly, note that we can quickly distinguish the equation of a hyperbola from that of a circle or
ellipse because the hyperbola formula involves a diﬀerence of squares where the circle and ellipse
formulas both involve the sum of squares.
(x − 2)2
Example 7.5.1. Graph the equation
= 1. Find the center, the lines which contain
the transverse and conjugate axes, the vertices, the foci and the equations of the asymptotes.
Solution. We ﬁrst see that this equation is given to us in the standard form of Equation 7.6.
Here x − h is x − 2 so h = 2, and y − k is y so k = 0. H...
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