4.6 Hyperbola (e > 1)
A hyperbola is the set of points P in the plane whose distances from a fixed point F are e times their distances from a fixed line d, where e > 1. We obtain a hyperbola in standard form if
the focus F lies on the x-axis, and has coordinates (ae, 0), where a > 0;
the directrix d is the line with equation x = a/e.
Let P (x, y) be an arbitrary point on the hyperbola, and let M be the foot of the perpendicular from P to the directrix. Since FP = e × PM, by the definition of the hyperbola, it follows that FP2 = e2 × PM2; that is,
Multiplying out the brackets, we obtain
x2 − 2aex + a2e2 + y2 = e2x2 − 2aex + a2,
which simplifies to the equation
x2(e2 − 1) − y2 = a2(e2 − 1),
Substituting b for , so that b2 = a2(e2 − 1), we obtain the standard form of the equation of the hyperbola
This equation is symmetric in x and in y, so that the hyperbola also has a second focus F′ at (−ae, 0), and a second directrix d′ with equation x = −a/e.
The hyperbola intersects the x-axis at the points (±a, 0). We call the line segment joining the points (±a, 0) the major axis or transverse axis of the hyperbola, and the line segment joining the points (0, ±b) the minor axis or conjugate axis of the hyperbola (this is not a chord of the hyperbola). The origin is the centre of this hyperbola.
Each point with coordinates (a sec t, b tan t) lies on the hyperbola, since
Note: In general, sec2t = 1 + tan2t.
Then, just as for the parabola, we can check that
gives a parametric representation of the hyperbola.
Note: An alternative parametrisation, using hyperbolic functions, is x = acosht, y = bsinht (t ∈ ).
Two other features of the shape of the hyperbola stand out.
First, the hyperbola consists of two separate curves or branches.
Secondly, the lines with equations y = ±bx/a divide the plane into two pairs of opposite sectors; the branches of the hyperbola lie in one pair. As x → ±∞, the branches of the hyperbola get closer and closer to these two lines. We call the lines y = ±bx/a the asymptotes of the hyperbola.
We summarise these facts as follows:
Hyperbola in standard form
A hyperbola in standard form has equation
It can also be described by the parametric equations
It has foci (±ae, 0) and directrices x = ±a/e; its major axis is the line segment joining the points (±a, 0), and its minor axis is the line segment joining the points (0, ±b).
Let P be a point , t ∈ , on the hyperbola with equation x2 − 2y2 = 1.
(a) Determine the foci F and F′ of the hyperbola.
(b) Determine the gradients of FP and F′P, when these lines are not parallel to the y-axis.
(c) Find the point P on the hyperbola, in the first quadrant, for which FP is perpendicular to F′P.
(a) This hyperbola is of the form with a = 1 and , so . If e denotes the eccentricity of the hyperbola, so that b2 = a2(e2 − 1), we have
it follows that , so
In the general case, the foci are (±ae, 0); it follows that here the foci are .
(b) Let F and F′ be and , respectively. (It does not matter which way round these are chosen.)
Then the gradient of FP is
where we know that , since FP is not parallel to the y-axis.
Similarly, the gradient of F′P is
where we know that , since F′P is not parallel to the y-axis.
(c) When FP is perpendicular to F′P, we have
We may rewrite this in the form
so 2sec2t − 3 + tan2t = 0.
Since sec2t = 1 + tan2t, it follows that 3 tan2t = 1.
Since we are looking for a point P in the first quadrant, we choose .
When , we have . Since we are looking for a point P in the first quadrant, we choose .
It follows that the required point P has coordinates .