Hyperbolas were introduced in
Introduction to Conic Sections,
as one of
several different curves (‘conic sections’) that are formed by intersecting a plane with an
infinite double cone.
Identifying Conics by the
Discriminant
introduced the general equation for any conic
section,
and gave conditions under which the graph would be a hyperbola.
In this current section, we present and explore the standard definition of a hyperbola.
This definition facilitates the derivation of standard equations for hyperbolas.
Recall that the notation ‘$\,d(P,Q)\,$’ denotes the distance between points $\,P\,$ and $\,Q\,.$
DEFINITION
hyperbola
A hyperbola is the set of points in a plane such that
the difference of the distances from
two fixed points is constant.
More precisely:
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![]() $\,P\,$ is a general point on the hyperbola. $|\,d(P,F_1) - d(P,F_2)| = \text{constant}$ $ \color{red}{\text{red (longer)}} - \color{blue}{\text{blue (shorter)}} = \text{constant} $ |
You can play with hyperbolas using the dynamic JSXGraph at right:
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foci: $\,F_1\,$ and $\,F_2\,,$ marked with ![]() vertices: marked in red $\,P\,$: a typical point on the hyperbola $\,Q\,$: a free point (can be moved anywhere) |
In the definition of hyperbola,
the hyperbola constant $\,k\,$ is required to be positive,
and strictly less than the distance between the two foci:
$$
\cssId{s106}{0 < k < d(F_1,F_2)}
$$
Why?
As shown below, other values of $\,k\,$ don't give anything that
a reasonable person would want to call a hyperbola!
$k < 0\,$: No Points in the Solution SetAbsolute value is always nonnegative. That is, $\,|x| \ge 0\,$ for all real numbers $\,x\,.$Therefore, if $\,k < 0\,,$ there are no points $\,P\,$ for which the hyperbola equation is true: $$ \cssId{s114}{\overbrace{\strut \,|d(P,F_1) - d(P,F_2)|}^{\,\ge\, 0\,} = \overbrace{\strut k}^{\lt \, 0} \qquad \text{ is always false }} $$ |
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$k = 0\,$: Perpendicular Bisector of Segment Between the FociThe equation $\,|x| = 0\,$ is equivalent to $\,x = 0\,,$ for all real numbers $\,x\,.$Thus, the following are equivalent: $$ \begin{gather} \cssId{s119}{|d(P,F_1) - d(P,F_2)| = 0}\cr\cr \cssId{s120}{d(P,F_1) - d(P,F_2) = 0}\cr\cr \cssId{s121}{d(P,F_1) = d(P,F_2)} \end{gather} $$ At right, $\,Q\,$ is the midpoint of the segment $\,\overline{F_1F_2}\,$ that connects the foci. When $\,k = 0\,,$ the set of points satisfying the hyperbola equation is the perpendicular bisector of $\,\overline{F_1F_2}\,.$ |
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A ‘TWO RAY’ HYPERBOLA: $\,k = d(F_1,F_2)\,$Suppose the hyperbola constant, $\,k\,,$ equals the distance between the foci.That is, $\,k = d(F_1,F_2)\,.$ In this case, the solution set to the equation $$ \cssId{s129}{|\color{green}{d(P,F_1)} - \color{red}{d(P,F_2)}| = k} $$ is two rays, with $\,F_1\,$ and $\,F_2\,$ as endpoints, lying along the major axis. This degenerate hyperbola is shown at right, in black. $\,P\,$ is a point on the degenerate hyperbola; it can be dragged from one ray to the other. |
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AN ‘EMPTY’ HYPERBOLA: $\,k > d(F_1,F_2)\,$Suppose $\,F_1\,$ and $\,F_2\,$ are distinct points in a plane, and let $\,k > d(F_1,F_2)\,.$Let $\,P\,$ be any point in the plane. Switch names (if needed) so that $\,P\,$ is closer to $\,F_1\,$ (or equidistant from both foci). This situation is illustrated at right. The shortest distance between any two points is a straight line. In particular (refer to the sketch at right), traveling from $\,P\,$ to $\,F_1\,$, and then from $\,F_1\,$ to $\,F_2\,$, must exceed (or equal) $\,\color{red}{d(P,F_2)}\,$: $$ \cssId{s150}{d(P,F_1) + d(F_1,F_2) \ge d(P,F_2)} $$ Re-arranging: $$ \cssId{s152}{d(P,F_2) - d(P,F_1) \le d(F_1,F_2)}\qquad \cssId{s153}{(*)} $$ We have: $$ \begin{alignat}{2} \cssId{s155}{|d(P,F_2) - d(P,F_1)|} &\ \cssId{s156}{=\ d(P,F_2) - d(P,F_1)} &\qquad &\cssId{s157}{\text{(for $\,x\ge 0\,,$ $\,|x| = x\,$)}}\cr &\ \cssId{s158}{\le\ d(F_1,F_2)}&&\cssId{s159}{\text{by (*)}}\cr &\ \cssId{s160}{< \ k}&&\cssId{s161}{\text{(by hypothesis)}} \end{alignat} $$ Consequently, there are no points $\,P\,$ for which the hyperbola equation is true: $$ \cssId{s163}{\overbrace{\strut \,|d(P,F_2) - d(P,F_1)|}^{\text{strictly less than $\,k\,$}} = k \qquad \text{ is always false }} $$ You might want to call this an empty hyperbola, an invisible hyperbola, or an imaginary hyperbola! There's nothing there! |
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Most people don't want to call any of these situations—the empty set, a line, or two rays—a hyperbola!
This is why these values for $\,k\,$ are not allowed in the definition of hyperbola.
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
IN PROGRESS |