7 Analytic Geometry
16. CONIC SECTIONS
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Solution
Put the equation in general form.
Axþ By þ C ¼ 2x y þ 3 ¼ 0 Use Eq. 7.49 withðx; yÞ ¼ ð0; 0Þ.
d¼jAx þ By þ Cjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2þ B2
p ¼ð2Þð0Þ þ ð1Þð0Þ þ 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ2þ ð1Þ2 q
¼ 3ffiffiffi p5
15. ANGLES BETWEEN GEOMETRIC FIGURES
The angle,, between various geometric figures is given by the following equations.
. between two lines in Ax + By + C = 0, y = mx + b, or direction angle formats:
¼ arctan A1B2 A2B1
A1A2þ B1B2
7:52
¼ arctan m2 m1 1þ m1m2
7:53
¼ jarctan m1 arctan m2j 7:54
¼ arccos L1L2þ M1M2þ N1N2 d1d2
7:55
¼ arccos cos1cos2þ cos 1cos2
þ cos 1cos2
!
7:56
If the lines are parallel, then = 0.
A1 A2¼B1
B2 7:57
m1¼ m2 7:58
1¼ 2; 1¼ 2; 1¼ 2 7:59
If the lines are perpendicular, then = 90.
A1A2¼ B1B2 7:60
m1¼ 1
m2 7:61
1þ 2¼ 1þ 2¼ 1þ 2 ¼ 90 7:62
. between two planes in Ai + Bj + C k = 0 format, the coefficients A, B, and C are the same as the coefficients for the normal vector. (See Eq. 7.37.) is equal to the angle between the two normal vectors.
cos ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijA1A2þ B1B2þ C1C2j A21þ B21þ C21
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A22þ B22þ C22
q 7:63
Example 7.5
Use Eq. 7.52, Eq. 7.53, and Eq. 7.54 to find the angle between the lines.
y¼ 0:577x þ 2 y¼ þ0:577x 5 Solution
Write both equations in general form.
0:577x y þ 2 ¼ 0 0:577x y 5 ¼ 0 (a) From Eq. 7.52,
¼ arctan A1B2 A2B1 A1A2þ B1B2
¼ arctan ð0:577Þð1Þ ð0:577Þð1Þ ð0:577Þð0:577Þ þ ð1Þð1Þ
¼ 60 (b) Use Eq. 7.53.
¼ arctan m2 m1 1þ m1m2
¼ arctan 0:577 ð0:577Þ 1þ ð0:577Þð0:577Þ
¼ 60 (c) Use Eq. 7.54.
¼ jarctan m1 arctan m2j
¼ jarctan 0:577 arctan 0:577j
¼ j30 30j
¼ 60
16. CONIC SECTIONS
A conic section is any one of several curves produced by passing a plane through a cone as shown in Fig. 7.10. If
is the angle between the vertical axis and the cutting plane and is the cone generating angle, Eq. 7.64 gives the eccentricity, , of the conic section. Values of the eccentricity are given in Fig. 7.10.
¼cos
cos 7:64
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All conic sections are described by second-degree polyno-mials (i.e., quadratic equations) of the following form.5
Ax2þ Bxy þ Cy2þ Dx þ Ey þ F ¼ 0 7:65
This is the general form, which allows the figure axes to be at any angle relative to the coordinate axes. The standard forms presented in the following sections per-tain to figures whose axes coincide with the coordinate axes, thereby eliminating certain terms of the general equation.
Figure 7.11 can be used to determine which conic section is described by the quadratic function. The quantity B2 4AC is known as the discriminant. Figure 7.11 determines only the type of conic section; it does not determine whether the conic section is degenerate (e.g., a circle with a negative radius).
Example 7.6
What geometric figures are described by the following equations?
(a) 4y2 12y + 16x + 41 = 0 (b) x2 10xy + y2+ x + y + 1 = 0 (c) x2þ 4y2þ 2x 8y þ 1 ¼ 0 (d) x2+ y2 6x + 8y + 20 = 0 Solution
(a) Referring to Fig. 7.11, B = 0 since there is no xy term, A = 0 since there is no x2 term, and AC ¼ ð0Þð4Þ ¼ 0.
This is a parabola.
(b) B6¼ 0; B2 4AC = (10)2 (4)(1)(1) = +96. This is a hyperbola.
(c) B = 0; A6¼ C; AC = (1)(4) = +4. This is an ellipse.
(d) B = 0; A = C; A = C = 1 (6¼ 0). This is a circle.
17. CIRCLE
The general form of the equation of a circle, as illus-trated in Fig. 7.12, is
Ax2þ Ay2þ Dx þ Ey þ F ¼ 0 7:66
5One or more straight lines are produced when the cutting plane passes through the cone’s vertex. Straight lines can be considered to be quadratic functions without second-degree terms.
Figure 7.10 Conic Sections Produced by Cutting Planes
circle perpendicularellipse
plane 90
(a) circle ( 90°)
0
(b) ellipse ( 90°) 0 1
(c) parabola ( = )
= 1 (d) hyperbolas (0 )
1
inclined plane
hyperbolas
= 0 shown parabola
Figure 7.11 Determining Conic Sections from Quadratic Equations
no yes
no yes
no yes
negative
positive negative
positive
ellipse parabola hyperbola
A C ?
ellipse parabola
hyperbola
A C 0 ?
line circle
AC
B 0 ?
B2 4AC zero
zero
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The center-radius form of the equation of a circle with radius r and center at (h, k) is
ðx hÞ2þ ðy kÞ2¼ r2 7:67 The two forms can be converted by use of Eq. 7.68 through Eq. 7.70.
If the right-hand side of Eq. 7.70 is positive, the figure is a circle. If it is zero, the circle shrinks to a point. If the right-hand side is negative, the figure is imaginary.
A degenerate circle is one in which the right-hand side is less than or equal to zero.
18. PARABOLA
A parabola is the locus of points equidistant from the focus (point F in Fig. 7.13) and a line called the direc-trix. A parabola is symmetric with respect to its para-bolic axis. The line normal to the parapara-bolic axis and passing through the focus is known as the latus rectum.
The eccentricity of a parabola is 1.
There are two common types of parabolas in the Cartesian plane—those that open right and left, and those that open up and down. Equation 7.65 is the gen-eral form of the equation of a parabola. With Eq. 7.71, the parabola points horizontally to the right if CD4 0 and to the left if CD 5 0. With Eq. 7.72, the parabola
The standard form of the equation of a parabola with vertex at ðh; kÞ, focus at ðh þ p; kÞ, and directrix at x = h p, and that opens to the right or left is given by Eq. 7.73. The parabola opens to the right (points to the left) if p4 0 and opens to the left
The standard form of the equation of a parabola with vertex at ðh; kÞ, focus at ðh; k þ pÞ, and directrix at y = k p, and that opens up or down is given by Eq. 7.75. The parabola opens up (points down) if p4 0 and opens down (points up) if p5 0.
ðx hÞ2¼ 4pðy kÞjopens vertically 7:75
x2¼ 4pyjvertex at origin 7:76
The general and vertex forms of the equations can be reconciled with Eq. 7.77 through Eq. 7.79. Whether the first or second forms of these equations are used depends on whether the parabola opens horizontally or vertically (i.e., whether A = 0 or C = 0), respectively.
h¼
E2 4CF
4C D ½opens horizontally
D
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19. ELLIPSE
An ellipse has two foci separated along its major axis by a distance 2c as shown in Fig. 7.14. The line perpendic-ular to the major axis passing through the center of the ellipse is the minor axis. The two lines passing through the foci perpendicular to the major axis are the latera recta. The distance between the two vertices is 2a. The ellipse is the locus of those points whose distances from the two foci add up to 2a. For each point P on the ellipse,
F1Pþ PF2¼ 2a 7:80
Equation 7.81 is the standard equation used for an ellipse with axes parallel to the coordinate axes, while Eq. 7.65 is the general form. F is not independent of A, C, D, and E for the ellipse.
Ax2þ Cy2þ Dx þ Ey þ F ¼ 0 AC> 0
A6¼ C
7:81
Equation 7.82 gives the standard form of the equation of an ellipse centered atðh; kÞ. Distances a and b are known as the semimajor distance and semiminor distance, respectively.
ðx hÞ2
a2 þðy kÞ2
b2 ¼ 1 7:82
The distance between the two foci is 2c.
2c¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 b2
p 7:83
The aspect ratio of the ellipse is aspect ratio¼a
b 7:84
The eccentricity,, of the ellipse is always less than 1. If the eccentricity is zero, the figure is a circle (another form of a degenerative ellipse).
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 b2 p
a < 1 7:85
The standard and center forms of the equations of an ellipse can be reconciled by using Eq. 7.86 through Eq. 7.89.
A hyperbola has two foci separated along its transverse axis (major axis) by a distance 2c, as shown in Fig. 7.15.
The line perpendicular to the transverse axis and mid-way between the foci is the conjugate axis (minor axis).
The distance between the two vertices is 2a. If a line is drawn parallel to the conjugate axis through either vertex, the distance between the points where it inter-sects the asymptotes is 2b. The hyperbola is the locus of those points whose distances from the two foci differ by 2a. For each point P on the hyperbola,
F2P PF1¼ 2a 7:90
Equation 7.91 is the standard equation of a hyperbola.
Coefficients A and C have opposite signs.
Ax2þ Cy2þ Dx þ Ey þ F ¼ 0jAC< 0 7:91
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Equation 7.92 gives the standard form of the equation of a hyperbola centered at (h, k) and opening to the left and right.
ðx hÞ2
a2 ðy kÞ2 b2 ¼ 1
opens horizontally
7:92
Equation 7.93 gives the standard form of the equation of a hyperbola that is centered at (h, k) and is opening up and down.
The distance between the two foci is 2c.
2c¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2þ b2
p 7:94
The eccentricity,, of the hyperbola is calculated from Eq. 7.95 and is always greater than 1.
¼c
The hyperbola is asymptotic to the lines given by Eq. 7.96 and Eq. 7.97.
For a rectangular (equilateral) hyperbola, the asymptotes are perpendicular, a = b, c¼ ffiffiffi
p2
a, and the eccentricity is
¼ ffiffiffi p2
. If the hyperbola is centered at the origin (i.e., h¼ k ¼ 0), then the equations are x2 y2 = a2 (opens horizontally) and y2 x2= a2(opens vertically).
If the asymptotes are the x- and y-axes, the equation of the hyperbola is simply
xy¼ ±a2
2 7:98
The general and center forms of the equations of a hyperbola can be reconciled by using Eq. 7.99 through Eq. 7.103. Whether the hyperbola opens left and right or up and down depends on whether M/A or M/C is positive, respectively, where M is defined by Eq. 7.99.
M¼D2
Equation 7.104 is the general equation of a sphere. The coefficient A cannot be zero.
Ax2þ Ay2þ Az2þ Bx þ Cy þ Dz þ E ¼ 0 7:104
Equation 7.105 gives the standard form of the equation of a sphere centered atðh; k; lÞ with radius r.
ðx hÞ2þ ðy kÞ2þ ðz lÞ2¼ r2 7:105 The general and center forms of the equations of a sphere can be reconciled by using Eq. 7.106 through Eq. 7.109.
A helix is a curve generated by a point moving on, around, and along a cylinder such that the distance the point moves parallel to the cylindrical axis is pro-portional to the angle of rotation about that axis. (See Fig. 7.16.) For a cylinder of radius r, Eq. 7.110 through Eq. 7.112 define the three-dimensional positions of points along the helix. The quantity 2pk is the pitch of the helix.
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