11.4 A Review of Conic Sections (Optional)

Three familiar families of curves—ellipses, hyperbolas, and parabolas—appear throughout mathematics and its applications. They are called conic sections because they are obtained as the intersection of a cone with a suitable plane (Figure 11.61). Our goal in this section is to derive equations for the conic sections from their geometric definitions as curves in the plane.

An ellipse is an oval-shaped curve [Figure 11.62] consisting of all points \(P\) such that the sum of the distances to two fixed points \(F_1\) and \(F_2\) is a constant \(K>0\): \begin{equation} PF_1 + PF_2 = K\tag{1} \end{equation}

The points \(F_1\) and \(F_2\) are called the foci (plural of “focus”) of the ellipse. Note that if the foci coincide, then Eq. (1) reduces to \(2PF_1=K\) and we obtain a circle of radius \(\frac12 K\) centered at \(F_1\).

We use the following terminology:

• The midpoint of \(\overline{F_1F_2}\) is the center of the ellipse.
• The line through the foci is the focal axis.
• The line through the center perpendicular to the focal axis is the conjugate axis.

The ellipse is said to be in standard position if the focal and conjugate axes are the \(x\)- and \(y\)-axes, as shown in Figure 11.62. In this case, the foci have coordinates \(F_1 = (c,0)\) and \(F_2=(-c,0)\) for some \(c > 0\). Let us prove that the equation of this ellipse has the particularly simple form \begin{equation} \left(\frac{x}a\right)^2 + \left(\frac{x}b\right)^2 = 1 \tag{2} \end{equation} where \(a=K/2\) and \(b=\sqrt{a^2-c^2}\).

By the distance formula, \(P = (x,y)\) lies on the ellipse in Figure 11.62 if \begin{equation} \label{12.conic.elldistance} PF_1+PF_2 = \sqrt{(x+c)^2+y^2}+\sqrt{(x-c)^2+y^2} = 2a\tag{3} \end{equation} Move the second term on the left over to the right and square both sides: \begin{align*} (x+c)^2+y^2 &= 4a^2 -4a\sqrt{(x-c)^2+y^2} + (x-c)^2+y^2\\ 4a\sqrt{(x-c)^2+y^2} &= 4a^2 + (x-c)^2 -(x+c)^2 = 4a^2-4cx \end{align*}

Now divide by 4, square, and simplify: \begin{align} a^2(x^2 - 2cx+c^2+y^2) &= a^4-2a^2cx + c^2x^2\notag\\ (a^2-c^2) x^2 +a^2y^2 &= a^4 -a^2c^2 = a^2(a^2-c^2)\notag\\ \frac{x^2}{a^2} +\frac{y^2}{a^2-c^2} &= 1\tag{4} \end{align}

This is Eq.(2) with \(b^2 = a^2-c^2\) as claimed.

The ellipse intersects the axes in four points \(A\), \(A'\), \(B\), \(B'\) called vertices. Vertices \(A\) and \(A'\) along the focal axis are called the focal vertices. Following common usage, the numbers \(a\) and \(b\) are referred to as the semimajor axis and the semiminor axis (even though they are numbers rather than axes).

To write down the equation of an ellipse with axes parallel to the \(x\)- and \(y\)-axes and center translated to the point \(C = (h, k)\), replace \(x\) by \(x-h\) and \(y\) by \(y-k\) in the equation (Figure 11.64): \[ \left(\frac{x-h}a\right)^2+\left(\frac{y-k}b\right)^2=1 \]

A hyperbola is the set of all points \(P\) such that the difference of the distances from \(P\) to two foci \(F_1\) and \(F_2\) is \(\pm K\): \[ {PF_1}-{PF_2}=\pm K\tag{6} \]

We assume that \(K\) is less than the distance \(F_1F_2\) between the foci (the hyperbola has no points if \(K>F_1F_2\)). Note that a hyperbola consists of two branches corresponding to the choices of sign \(\pm\) (Figure 11.65).

As before, the midpoint of \(\overline{F_1F_2}\) is the center of the hyperbola, the line through \(F_1\) and \(F_2\) is called the focal axis, and the line through the center perpendicular to the focal axis is called the conjugate axis. The vertices are the points where the focal axis intersects the hyperbola; they are labeled \(A\) and \(A'\) in Figure 11.65. The hyperbola is said to be in standard position when the focal and conjugate axes are the \(x\)- and \(y\)-axes as in Figure 11.66. The next theorem can be verified in much the same way as Theorem 1.

A hyperbola has two asymptotes \({y=\pm\frac{b}ax}\) which are, we claim, diagonals of the rectangle whose sides pass through \((\pm a,0)\) and \((0,\pm b)\) as in Figure 11.66.

To prove this, consider a point \((x,y)\) on the hyperbola in the first quadrant. By Eq.(7), \[ y = \sqrt{\frac{b^2}{a^2}x^2-b^2} = \frac{b}a\sqrt{x^2-a^2} \]

The following limit shows that a point \((x,y)\) on the hyperbola approaches the line \({y=\frac{b}ax}\) as \(x\to\infty\): \begin{align*} \lim_{x\to\infty}\left(y-\frac{b}ax\right) &=\frac{b}a\lim_{x\to\infty}\left(\sqrt{x^2-a^2}- x\right)\\[-2pt] &=\frac{b}a\lim_{x\to\infty}\left(\sqrt{x^2-a^2}- x\right)\left(\frac{\sqrt{x^2-a^2}+ x}{\sqrt{x^2-a^2}+ x}\right)\\[-2pt] &=\frac{b}a\lim_{x\to\infty} \left(\frac{-a^2}{\sqrt{x^2-a^2}+ x}\right) = 0 \end{align*}

The asymptotic behavior in the remaining quadrants is similar.

Unlike the ellipse and hyperbola, which are defined in terms of two foci, a parabola is the set of points \(P\) equidistant from a focus \(F\) and a line \(\mathcal D\) called the directrix: \begin{equation} {PF} = {P\mathcal D}\tag{8} \end{equation}

Here, when we speak of the distance from a point \(P\) to a line \(\mathcal D\), we mean the distance from \(P\) to the point \(Q\) on \(\mathcal D\) closest to \(P\), obtained by dropping a perpendicular from \(P\) to \(\mathcal D\) (Figure 11.68). We denote this distance by \({P\mathcal D}\).

The line through the focus \(F\) perpendicular to \(\mathcal D\) is called the axis of the parabola. The vertex is the point where the parabola intersects its axis. We say that the parabola is in standard position if, for some \(c\), the focus is \(F = (0,c)\) and the directrix is \(y = -c\), as shown in Figure 11.68. We verify in Exercise 73 that the vertex is then located at the origin and the equation of the parabola is \({y = {x^2}/{4c}}\). If \(c \lt 0\), then the parabola opens downward.

Eccentricity

Some ellipses are flatter than others, just as some hyperbolas are steeper. The “shape” of a conic section is measured by a number \(e\) called the eccentricity. For an ellipse or hyperbola, \[ e = \frac{\textrm{distance betweeen foci}}{\textrm{distance between vertices on focal axis}} \]

A parabola is defined to have eccentricity \(e = 1\).

How does eccentricity determine the shape of a conic [Figure 11.70]? Consider the ratio \(b/a\) of the semiminor axis to the semimajor axis of an ellipse. The ellipse is nearly circular if \(b/a\) is close to \(1\), whereas it is elongated and flat if \(b/a\) is small. Now \[ \frac{b}a = \frac{\sqrt{a^2-c^2}}a = \sqrt{1-\frac{c^2}{a^2}} =\sqrt{1-e^2} \]

This shows that \(b/a\) gets smaller (and the ellipse get flatter) as \(e\to 1\) [Figure 11.70]. The “roundest” ellipse is the circle, with \(e=0\).

Similarly, for a hyperbola, \[\frac{b}{a}=\sqrt{1+e^2}\]

The ratios \(\pm b/a\) are the slopes of the asymptotes, so the asymptotes get steeper as \(e\to\infty\) [Figure 11.70].

Eccentricity can be used to give a unified focus-directrix definition of the conic sections. Given a point \(F\) (the focus), a line \(\mathcal D\) (the directrix), and a number \(e \gt 0\), we consider the set of all points \(P\) such that \begin{equation} \boxed{\bbox[#fef7e5,5pt]{PF = eP\mathcal D}}\tag{10} \end{equation}

For \(e=1\), this is our definition of a parabola. According to the next theorem, Eq.(10) defines a conic section of eccentricity \(e\) for all \(e \gt 0\) (Figure 11.71 and Figure 11.72). Note, however, that there is no focus-directrix definition for circles (\(e=0\)).

In Section 13.8, we discuss the famous law of Johannes Kepler stating that the orbit of a planet around the sun is an ellipse with one focus at the sun. In this discussion, we will need to write the equation of an ellipse in polar coordinates. To derive the polar equations of the conic sections, it is convenient to use the focus-directrix definition with focus \(F\) at the origin \(O\) and vertical line \(x=d\) as directrix \(\mathcal D\) (Figure 11.75).

Note from the figure that if \(P=(r,\theta)\), then \[ {PF} = r,\qquad {P\mathcal D} = d - r\cos\theta \]

Thus the focus-directrix equation of the ellipse \({{PF}=e{P\mathcal D}}\) becomes \(r =e(d-r\cos\theta)\), or \(r(1+ e\cos\theta) =ed\). This proves the following result, which is also valid for the hyperbola and parabola (see Exercise 67).

Reflective Properties of Conic Sections

The conic sections have numerous geometric properties. Especially important are the reflective properties, which are used in optics and communications (for example, in antenna and telescope design; Figure 11.76).

We describe these properties here briefly without proof (but see Exercises 68–70 and Exercise 71 for proofs of the reflective property of ellipses).

• Ellipse: The segments \(F_1P\) and \(F_2P\) make equal angles with the tangent line at a point \(P\) on the ellipse. Therefore, a beam of light originating at focus \(F_1\) is reflected off the ellipse toward the second focus \(F_2\) [Figure 11.77]. See also Figure 11.78.
• Hyperbola: The tangent line at a point \(P\) on the hyperbola bisects the angle formed by the segments \(F_1P\) and \(F_2P\). Therefore, a beam of light directed toward \(F_2\) is reflected off the hyperbola toward the second focus \(F_1\) [Figure 11.77].
• Parabola: The segment \(FP\) and the line through \(P\) parallel to the axis make equal angles with the tangent line at a point \(P\) on the parabola [Figure 11.77]. Therefore, a beam of light approaching \(P\) from above in the axial direction is reflected off the parabola toward the focus \(F\).

General Equations of Degree 2

The equations of the standard conic sections are special cases of the general equation of degree 2 in \(x\) and \(y\): \begin{equation} \boxed{\bbox[#fef7e5,5pt]{ax^2+bxy+cy^2 + dx+ey+f = 0}}\tag{12} \end{equation}

Here \(a\), \(b\), \(e\), \(d\), \(e\), \(f\) are constants with \(a\), \(b\), \(c\) not all zero. It turns out that this general equation of degree 2 does not give rise to any new types of curves. Apart from certain “degenerate cases,” Eq.(12) defines a conic section that is not necessarily in standard position: It need not be centered at the origin, and its focal and conjugate axes may be rotated relative to the coordinate axes. For example, the equation \[ 6x^2-8xy+8y^2-12x-24y+38=0 \] defines an ellipse with center at \((3,3)\) whose axes are rotated (Figure 11.79).

We say that Eq.(12) is degenerate if the set of solutions is a pair of intersecting lines, a pair of parallel lines, a single line, a point, or the empty set. For example:

• \(x^2-y^2= 0\) defines a pair of intersecting lines \(y=x\) and \(y=-x\).
• \(x^2-x= 0\) defines a pair of parallel lines \(x=0\) and \(x=1\).
• \(x^2 =0\) defines a single line (the \(y\)-axis).
• \(x^2+y^2= 0\) has just one solution \((0,0)\).
• \(x^2+y^2=-1\) has no solutions.

Now assume that Eq.(12) is nondegenerate. The term \(bxy\) is called the cross term. When the cross term is zero (that is, when \(b=0\)), we can “complete the square” to show that Eq.(12) defines a translate of the conic in standard position. In other words, the axes of the conic are parallel to the coordinate axes. This is illustrated in the next example.

When the cross term \(bxy\) is nonzero, Eq.(12) defines a conic whose axes are rotated relative to the coordinate axes.

The marginal note describes how this may be verified in general. We illustrate with the following example.

We conclude our discussion of conics by stating the Discriminant Test. Suppose that the equation \[ ax^2+bxy+cy^2 + dx+ey+f = 0 \]

is nondegenerate and thus defines a conic section. According to the Discriminant Test, the type of conic is determined by the discriminant \(D\): \[\boxed{\bbox[#fef7e5,5pt]{D = b^2-4ac}} \]

We have the following cases:

• \(D \lt 0\): Ellipse or circle
• \(D \gt 0\): Hyperbola
• \(D=0\): Parabola

For example, the discriminant of the equation \(2xy=1\) is \[ D = b^2-4ac=2^2-0 = 4 >0 \]

According to the Discriminant Test, \(2xy=1\) defines a hyperbola. This agrees with our conclusion in Example 8.

Summary

• An ellipse with foci \(F_1\) and \(F_2\) is the set of points \(P\) such that \({{PF_1} + {PF_2} = K}\), where \(K\) is a constant such that \(K \gt F_1F_2\). The equation in standard position is \[ \left(\frac{x}a\right)^2+\left(\frac{y}b\right)^2=1 \]

  The vertices of the ellipse are \((\pm a, 0)\) and \((0, \pm b)\).

  	Focal axis	Foci	Focal vertices
  \(a \gt b\)	\(x\)-axis	\((\pm c, 0)\) with \(c=\sqrt{a^2-b^2}\)	\((\pm a,0)\)
  \(a \lt b\)	\(y\)-axis	\((0,\pm c)\) with \(c=\sqrt{b^2-a^2}\)	\((0,\pm b)\)

  Eccentricity: \({e=\tfrac{c}a}\) \((0\le e \lt 1)\). Directrix: \({x=\tfrac{a}e}\) (if \(a \gt b\)).

• A hyperbola with foci \(F_1\) and \(F_2\) is the set of points \(P\) such that \[ {PF_1} - {PF_2} = \pm K \]

  where \(K\) is a constant such that \(0 \lt K \lt F_1F_2\). The equation in standard position is \[ \left(\frac{x}a\right)^2-\left(\frac{y}b\right)^2=1 \]

  Focal axis	Foci	Vertices	Asymptotes
  \(x\)-axis	\((\pm c,0)\) with \(c=\sqrt{a^2+b^2}\)	\((\pm a,0)\)	\(y=\pm\dfrac{b}{a}x\)

  Eccentricity: \({e=\tfrac{c}a}\) \((e \gt 1)\). Directrix: \({x=\tfrac{a}e}\).

• A parabola with focus \(F\) and directrix \(\mathcal D\) is the set of points \(P\) such that \({{PF}={P\mathcal D}}\). The equation in standard position is \[ y=\frac1{4c}x^2 \]

  Focus \(F=(0,c)\), directrix \(y = - c\), and vertex at the origin \((0,0)\).

• Focus-directrix definition of conic with focus \(F\) and directrix \(\mathcal D\): \(PF = eP\mathcal D\).
• To translate a conic section \(h\) units horizontally and \(k\) units vertically, replace \(x\) by \(x-h\) and \(y\) by \(y-k\) in the equation.
• Polar equation of conic of eccentricity \(e \gt 0\), focus at the origin, directrix \(x = d\): \[ r = \frac{ed}{1 + e\cos\theta} \]