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the slopes of the asymptotes are ± a . Since a = 5 and the slope of the line y = 2x is 2, we have
that 5 = 2, so b = 10. Hence, b2 = 100 and our ﬁnal answer is
25 100 As with the other conic sections, an equation whose graph is a hyperbola may be given in a form
other than the standard forms in Equations 7.6 or 7.7. In those cases, as with conic sections which
have come before, we will need to massage the given equation into one of the forms in Equations
7.6 or 7.7.
To Write the Equation of a Hyperbola in Standard Form
1. Group the same variables together on one side of the equation and position the constant on
the other side
2. Complete the square in both variables as needed
3. Divide both sides by the constant term so that the constant on the other side of the equation
Example 7.5.3. Consider the equation 9y 2 − x2 − 6x = 10. Put this equation in to standard form
and graph. Find the center, the lines which contain the transverse and conjugate axes, the vertices,
the foci, and the equations of the asymptotes.
Solution. We need only complete the square on the x, and then divide, if necessary, to get the...
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