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Math 137 – Conic Sections – Section 11.3 Hyperbolas
Terminology and Symbols dealing with Hyperbolas The definition of a hyperbola is similar to that of an ellipse. The only change is that instead of using the sum of distances from two fixed points, we use the difference. Referring to Figure 1, we see that a point P ( x , y ) is on the hyperbola if and only if either of the following is true: (1) d ( P , F ) – d ( P , F ) = 2 a or (2) d ( P , F ) – d ( P , F ) = 2 a Using the distance formula to find d ( P , F ) and d ( P , F ), we obtain an equation of the hyperbola: Employing the type of simplification procedure, we can rewrite the preceding equation as
Finally, if we let b^2 = c^2 – a^2 with b > 0 in the preceding equation, we obtain We have shown that the coordinates of every point ( x , y ) on the hyperbola satisfy the equation ( x^2 / a^2 ) – ( y^2 / b^2 ) = 1. Applying tests for symmetry, we see that the hyperbola is symmetric with respect to both axes and the origin. We may find the x - intercepts of the hyperbola by letting y = 0 in the equation. Doing so gives us x^2 / a^2 = 1, or x^2 = a^2 , and consequently the x - intercepts are a and – a. The corresponding points V ( a , 0) and V (– a , 0) on the graph are called the vertices of the hyperbola.
A convenient way to sketch the asymptotes is to first plot the vertices V ( a , 0), V (– a , 0) and the points W (0, b ), W (0, – b ) If vertical and horizontal lines are drawn through these endpoints of the transverse and conjugate axes, respectively, then the diagonals of the resulting auxiliary rectangle have slopes b / a and – b / a. Hence, by extending these diagonals we obtain the asymptotes y = ( b / a ) x. The hyperbola is then sketched, using the asymptotes as guides. The two parts that make up the hyperbola are called the right branch and the left branch of the hyperbola. Similarly, if we take the foci on the y - axis, we obtain the equation In this case, the vertices of the hyperbola are (0, a ) and the endpoints of the conjugate axis are ( b , 0), as shown in Figure 3.
The asymptotes are y = ( a / b ) x ( not y = ( b / a ) x , as in the previous case), and we now refer to the two parts that make up the hyperbola as the upper branch and the lower branch of the hyperbola. The preceding discussion may be summarized as follows.
Example 4 : Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. foci F (±6, 0), vertices V (±3, 0) As was the case for ellipses, we may use translations to help sketch hyperbolas that have centers at some point ( h , k ) ≠ (0, 0). Example 5: Sketch the graph of the equation 9 x^2 – 4 y^2 – 54 x – 16 y + 29 = 0.
