364CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

through rotating the xy-coordinate system through the angle θ. Then the given second degree equation

ax2 + bxy + cy2 + dx + ey + f = 0

becomes

a0x02 + c0y02 + d0x + e0y + f0 = 0

where

a0 = a cos2 θ + b cos θ sin θ + c sin2 θ c0 = a sin2 θ − b sin θ cos θ + c cos2 θ d0 = d cos θ + e sin θ

e0 = −d sin θ + e cos θ f0 = f

Furthermore, the given second degree equation represents

(i)an ellipse, a circle, a point or no graph if b2 − 4ac < 0;

(ii)a hyperbolic or a pair of intersecting lines if b2 − 4ac > 0;

(iii)a parabola, a line, a pair of parallel lines, or else no graph if b2 −4ac = 0.

9.5Polar Coordinates

Definition 9.5.1 Each point P (x, y) in the xy-coordinate plane is assigned the polar coordinates (r, θ) that satisfy the following relations:

x2 + y2 = r2, y = r cos θ, y = r sin θ.

The origin is called the pole and the positive x-axis is called the polar axis. The number r is called the radial coordinate and the angle θ is called the angular coordinates. The polar coordinates of a point are not unique as the rectangular coordinates are. In particular,

(r, θ) ≡ (r, θ + 2nπ) ≡ (−r, θ + (2m + 1)π)

where n and m are any integers. There does exist a unique polar representation (r, θ) if r ≥ 0 and 0 ≤ θ < 2π.

9.6Graphs in Polar Coordinates

Theorem 9.6.1 A curve in polar coordinates is symmetric about the

(a)x-axis if (r, θ) and (r, −θ) both lie on the curve;

(b)y-axis if (r, θ) and (r, π − θ) both lie on the curve;

(c)origin if (r, θ), (r, θ + π) and (−r, θ) all lie on the curve.

Theorem 9.6.2 Let e be a positive number. Let a fixed point F be called the focus and a fixed line, not passing through the focus, be called a directrix. If P is a point in the plane, let P F stand for the distance between P and the focus F and let P D stand for the distance between P and the directrix. Then the locus of all points P such that P F = eP D is a conic section representing

(a)an ellipse if 0 < e < 1;

(b)a parabola if e = 1;

(c)a hyperbola if e > 1;

The number e is called the eccentricity of the conic.

In particular an equation of the form

ek

r = 1 ± e cos θ

represents a conic with eccentricity e, a focus at the pole (origin), and a directrix perpendicular to the polar axis and k units to the right of the pole, in the case of + sign, and k units to the left of the pole, in the case of − sign.

Also, an equation of the form

ek

r = 1 ± e sin θ

represents a conic with eccentricity e, a focus at the pole, and a directrix parallel to the polar axis and k units above the pole, in the case of + sign, and k units below the pole, in the case of − sign.

366CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

9.7Areas in Polar Coordinates

Theorem 9.7.1 Let r = f(θ) be a curve in polar coordinates such that f is continuous and nonnegative for all α ≤ θ ≤ β where α ≤ β ≤ 2π + α. Then the area A bounded by the curves r = f(θ), θ = α and θ = β is given by

Theorem 9.7.2 Let r = f(θ) be a curve in polar coordinates such that f and f0 are continuous for α ≤ θ ≤ β, and there is no overlapping, the arc length L of the curve from θ = α to θ = β is given by

9.8 Parametric Equations

Definition 9.8.1 A parametrized curve C in the xy-plane has the form

C = {(x, y) : x = f(t), y = g(t), t I}

for some interval I, finite or infinite.

The functions f and g are called the coordinate functions and the variable t is called the parameter.

Theorem 9.8.1 Suppose that x = f(t), y = g(t) are the parametric equations of a curve C. If f0(t) and g0(t) both exist and f0(t) 6= 0, then

At a point P0(f(t0), g(t0)), the equation of

(a) the tangent line is

g0(t0)

y − g(t0) = f0(t0) (x − f(t0))

(b) the normal line is

f0(t0)

y − g(t0) = −g0(t0) (x − f(t0))

provided g0(t0) 6= 0 and f0(t0) 6= 0.

Theorem 9.8.2 Let C = {(x, y) : x = f(t), y = g(t), a ≤ t ≤ b} where f0(t) and g0(t) are continuous on [a, b]. Then the arc length L of C is given by

Z b

L = [(f0(t))2 + (g0(t))2]1/2dt

Theorem 9.8.3 Let C = {(x, y) : x = f(t), y = g(t), a ≤ t ≤ b}, where f0(t) and g0(t) are continuous on [a, b].

(a)If C lies in the upper half plane or the lower half plane and there is no overlapping, then the surface area generated by revolving C around the x-axis is given by

Z b

p

2πg(t) (f0(t))2 + (g0(t))2 dt.

a

(b)If 0 ≤ f(t) on [a, b], (or f(t) ≤ 0 on [a, b]) and there is no overlapping, then the surface area generated by revolving C around the y-axis is

Z b

p

2πf(t) (f0(t))2 + (g0(t))2 dt.

a

368CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

Definition 9.8.2 Let C = {(x(t), y(t)) : a ≤ t ≤ b} for some interval I. Suppose that x0(t), y0(t), x00(t) and y00(t) are continuous on I.

(a) The arc length s(t) is defined by

Z t

s(t) = [(x0(t))2 + (y0(t))2]1/2dt.

a

(b) The angle of inclination, φ, of the tangent line to the curve C is defined

(d) The radius of curvature, R, is defined by

R(t) = κ(1t).