9.3. The Parabola
We met parabolas in Chapters 1 and 2. The parabola is in many ways the most interesting of the three conic sections. Parabolas lie midway between ellipses and hyperbolas. Like hyperbolas, parabolas are open curves, but unlike them and like ellipses, they only have one component. Unlike ellipses or hyperbolas, all parabolas are similar, i.e. they all have the same shape. Aslo, as we will see, parabolas have many practical and physical applications.
A parabola is defined as the locus of points in a plane equidistant from a fixed point F, known as the focus, and a line L in the plane not containing F, known as the directrix. Note that unlike ellipses and hyperbolas, parabolas have only one focus. A graph of a parabola is shown below, along with its focus F, directrix L (dotted line), points P1, P2, and P3 lying on the parabola, and points Q1, Q2, and Q3lying on the directrix along vertical lines (perpendicular to the directrix) containing P1, P2, and P3 respectively. Note that for each of these three points Pi shown in the figure, we have
- (9.3.1) d(F, Pi) = d(Pi, Qi) = d(Pi,L),
the last equality holding by definition of the distance from a point to a line. Two solid lines are also shown. These are the axes of the parabola. The major axis or axis of symmetry is the line (vertical line shown in the figure) containing the focus and perpendicular to the directrix. The point of intersection of the parabola and the major axis (not shown in the figure) is called the vertex of the parabola. The other solid line (horizontal in the figure) perpendicular to the major axis and also containing the vertex, is the minor axis. The minor axis is the tangent line to the parabola at the vertex.
Figure 9.3.2: The Parabola
(Image taken from Wikipedia)
The most well-known parabola is the one given by the equation y = x2. We encountered this parabola in Chapter 1. This parabola has its vertex at the origin, the y-axis as its major axis and the x-axis as its minor axis. More generally, as we have mentioned in Chapter 2, the graphs of all quadratic polynomials, i.e. polynomials of the form y = f(x) = ax2 + bx + c, where a, b, and c are constants with a nonzero, are parabolas. All these parabolas have vertical major axes and horizontal minor axes.
So what does the most general equation of a parabola look like? In this section, we only consider parabolas with vertical and horizontal axes. As it turns out, the most general equation of a parabola with horizontal major axis and vertex at the origin has the following form:
- (9.3.3) y2 = 4px.
Here, p is a nonzero constant whose absolute value is equal to the distance from the focus to the vertex, which is also the distance from the vertex to the directrix. If p is positive, then the parabola opens up to the right; otherwise it opens up to the left. Similarly, the most general equation of a parabola with vertical major axis and vertex at the origin has the form
- (9.3.4) x2 = 4py.
Here, if p is positive, then the parabola opens up to the north (upward) and if p is negative, the parabola opens up to the south (downward).
Example 1: Find the vertex, focus, and directrix of the parabola y = x2.
Solution: Note that this parabola has the form of Equation (9.3.4) with p = 1/4. Thus, we see that it has vertex (0, 0). The focus is located a distance p = 1/4 from the vertex along the major axis, which is vertical. Thus, the focus has coordinates (0, 1/4). The directrix is located a distance 1/4 from the vertex in the opposite direction as the focus. Thus, it contains the point (0, 1/4) and is horizontal (perpendicular to the major axis and parallel to the minor axis). Therefore, the equation of the directrix is y = -1/4.
So what about parabolas whose vertex is not at the origin? These parabolas are displaced from the ones we just considered by arbitary distances x0 and y0 in the x and y directions, respectively. Therefore, they either have the form
- (9.3.5) (y - y0)2 = 4p(x - x0)
- (9.3.6) (x - x0)2 = 4p(y - y0).
In the second case, it is a straightforward but somewhat tedious exercise, which we leave to the reader, to verify that Equation (9.3.6) is equivalent to the familiar quadratic form
- (9.3.7) y = ax2 + bx + c
with a, b, and c given by the following formulas:
- (9.3.8a) p = 1/4a
- (9.3.8b) x0 = -b/2a
- (9.3.8c) y0 = (4ac - b2) / 4a = -D/4a
where D = b2 - 4ac is the discriminant of the quadratic polynomial ax2 + bx + c. Note that if a is positive, then y0 is negative if and only if D is positive. This makes sense in light of what we know about quadratic equations, since y0 negative implies that the parabola dips below the x-axis, whence the polynomial f(x) = ax2 + bx + c has two roots, namely the values of x corresponding to the two intersection points of the parabola with the x-axis. On the other hand, if D is negative, then y0 is positive, implying that f(x) has no real roots, since the parabola stays above the x-axis. In the intermediate case in which D = 0, y0 must also be zero, implying that the vertex of the parabola lies on the x-axis, whence f(x) has exactly one root, namely x0 = -b/2a.
It is easier to go from p, x0, and y0 to a, b, and c, since to do so, we merely need to expand both sides of Equation (9.3.6). The result, which we also leave as an exercise, is the following set of formulas:
- (9.3.9a) a = 1/4p
- (9.3.9b) b = -x0/2p
- (9.3.9c) c = y0 + (x02/4p)
Example 2: Find the vertex, focus, and directrix of the parabola given by the equation y = x2 + x + 1.
Solution: Using Equations (9.3.8) with a = b = c = 1, we find that the parabola has the form (x - x0)2 = 4p(y - y0) with p = 1/4, x0 = -1/2, and y0 = 3/4. (Alternatively, we could solve for p, x0, and y0 by completing the square on the right side of the given equation for the parabola.) Thus, the vertex of the parabola is at (x0, y0) = (-1/2, 3/4). The focus lies a distance 1/4 above the vertex, whence it is at (-1/2, 1). The directrix is the horizontal line lying a distance 1/4 below the vertex, whence it is given by the equation y = 1/2.
Example 3: A parabola has focus (2, 7) and directrix x = 8. Find its vertex and its equation.
Solution: The focus lies six units to the left of the directrix, whence p = -3. Since the directrix is vertical, the parabola has the form (9.3.5). The vertex must lie three units to the right of the focus, whence it lies at (5, 7), implying x0= 5 and y0 = 7. Plugging these three values into (9.3.5), we find the parabola has the equation (y - 7)2 = -12(x - 5).
Like ellipses, parabolas have a very useful reflective property. In fact, the reflective property of the parabola is even more useful than that of the ellipse. The property is as follows: Consider a set of three vertical lines shown in the figure below. These lines are all perpendicular to the dotted line L shown, which is an arbitary line parallel to the minor axis and lying further from the vertex than the focus. For i = 1, 2, or 3, we let Pi denote the point of intersection of each of these lines with the parabola and Pi denote the point of intersection of each of these lines with the line L. Now consider the line segments FPi and PiQi. It turns out that the angles that each of these line segments make with the parabola are equal. What this means in terms of reflection of light or sound waves is that a set of plane waves approaching the parabola along the major axes will all converge on F after they are reflected off the parabola. Similarly, a set of waves from a point source at F will all emerge as plane waves moving upward after being relfected by the parabola.
Figure 9.3.10: Reflective Property of the Parabola
(Image taken from Wikipedia)
The reflective property of the parabola has numerous practical applications. Since light is a wave, if a light source may be placed at the focus of a paraboloid (the surface formed by rotating a parabola about its major axis), the result will be a focused beam of light emerging outward along the direction of the axis. This is how flashlights, headlights, and searchlights work. The opposite is also true. A parallel beam of light or any other kind of wave incident on a paraboloid will converge on the focus after being reflected. This property is used in reflecting telescopes and satellite dishes.
Besides their reflective property, parabolas have many other useful applications. Neglecting air resistance, the projectile of a body in free-fall near the surface of the earth is a parabola. This is due to the law of falling bodies, which we touched on in Section 1.10. A falling body has two components of its motion, one horizontal and one vertical. In the horizontal direction, its velocity is constant, but in the vertical direction, it accelerates toward the ground at a rate of 9.8 meters per second per second. The resulting curve is a downward-pointing parabola.
Another useful application of parabolas is in the design of suspension bridges. As it turns out, the shape of the cable of a completed suspension bridge is a parabola, although before the road is in place, it has a different shape, known as a catenary. Similar to a parabola but opening up somewhat faster, a catenary is the shape of a dangling chain.