Lecture Note Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 10.2 Plane Curves and Parametric Equations 10.3 Parametric Equations and Calculus 10.4 Polar Coordinates and Polar Graphs 10.5 Area and Arc Length in Polar Coordinates 10.6 Polar Equations of Conics and Kepler’s Laws Lecture Note Chapter 10 Conics, Parametric Equations, and Polar Coordinates 10.1 Conics and Calculus 10.2 Plane Curves and Parametric Equations 10.3 Parametric Equations and Calculus 10.4 Polar Coordinates and Polar Graphs 10.5 Area and Arc Length in Polar Coordinates 10.6 Polar Equations of Conics and Kepler’s Laws Lecture Note 10.1 Conics and Calculus

Geometric Definitions of Conic Sections and Their Standard Equations Each conic section (or simply conic) can be described as the intersection of a plane and a double-napped cone. There are seven different possible intersections.

Three Others When the intersecting plane passes through the vertex of the cone.

Four Basic Conics For four basic conics, the intersecting plane does not pass through the vertex of the cone. Lecture Note 10.1 Conics and Calculus

Geometric Definitions of Conic Sections and Their Standard Equations Parabolas, ellipses, and hyperbolas

1. The paths of the planets around the sun (ellipses) 2. Parabolic mirrors are used to converge light beams at the focus of the parabola 3. Parabolic microphones perform a similar function with sound waves 4. Parabola is used in the design of car headlights and in spotlights 5. Trajectory of objects thrown or shot near the earth’s surface will follow a parabolic path 6. Hyperbolas are used in a navigation system known as LORAN(long range navigation) General second-degree equation 7. Hyperbolic as well as parabolic 2 2 퐴푥 + 퐵푥푦 + 퐶푦 + 퐷푥 + 퐸푦 + 퐹 = 0 mirrors and lenses are used in systems of telescopes. Lecture Note 10.1 Conics and Calculus One of approaches to define the conics is that the conics is defined as a locus (collection) of points satisfying a certain geometric property. For example, a circle can be defined as the collection of all points (푥, 푦) that are equidistant from a fixed point (ℎ, 푘). This locus definition easily produces the standard equation of a circle 푥 − ℎ 2 + 푦 − 푘 2 = 푟2. Lecture Note 10.1 Conics and Calculus

Definition of Parabolas A parabola is a set of points in a plane that are equidistance from a fixed point 퐹 (called the focus) and a fixed line (called the directrix). The point halfway between the focus and the directrix lies on the parabola and it is called the vertex. The line through the focus perpendicular to the directrix is called the axis of the parabola.

(0, 푝) Standard Equation 퐹 (focus) (푥, 푦) An equation of the parabola with focus (0, 푝) and directrix 푦 = −푝 is (0,0) 푥2 = 4푝푦 (See Figure) directrix 푦 = −푝 vertex axis Lecture Note 10.1 Conics and Calculus

(0, 푝) (푥, 푦) Standard Equation 퐹 (focus) An equation of the parabola with focus (0, 푝) and directrix 푦 = −푝 is (0,0) 푥2 = 4푝푦 (See Figure) directrix 푦 = −푝 vertex axis

vertex Standard Equation (푝, 0) An equation of the parabola with focus (푝, 0) and directrix (0,0) 퐹 (focus) axis 푥 = −푝 is 푦2 = 4푝푥

(푥, 푦) directrix 푥 = −푝 Lecture Note 10.1 Conics and Calculus Theorem 10.1 (Standard Equation of a Parabola) The standard form of the equation of a parabola with vertex (ℎ, 푘) and directrix 푦 = 푘 − 푝 is 푥 − ℎ 2 = 4푝 푦 − 푘 푉푒푟푡𝑖푐푎푙 푎푥𝑖푠 For directrix 푥 = ℎ − 푝, the equation is 푦 − 푘 2 = 4푝 푥 − ℎ 퐻표푟𝑖푧표푛푡푎푙 푎푥𝑖푠 The focus lies on the axis 푝 units (directed distance) from the vertex. The coordinates of the focus are as follows ℎ, 푘 + 푝 푉푒푟푡𝑖푐푎푙 푎푥𝑖푠 ℎ + 푝, 푘 퐻표푟𝑖푧표푛푡푎푙 푎푥𝑖푠

Example 1. Find the focus of the parabola 1 1 푦 = − 푥 − 푥2 2 2 Lecture Note 10.1 Conics and Calculus

The Latus Rectum and Graphing Parabolas The latus rectum of a parabola is a line segment that passes through its focus, is parallel to its directrix, and has its endpoints on the parabola. The length of the latus rectum for the graphs of 푦2 = 4푝푥 and 푥2 = 4푝푦 is |4푝|.

푝 푝 푝 푝 2푝

4푝

Example 2. Find the length of the latus rectum of the parabola 푥2 = 4푝푦. Then find the length of the parabolic arc intercepted by the latus rectum. Lecture Note 10.1 Conics and Calculus Theorem 10.2 (Reflective Property of a Parabola) Let 푃 be a point on a parabola. The tangent line to the parabola at point 푃 makes equal angles with the following two lines 1. The line passing through 푃 and the focus. 2. The line passing through 푃 parallel to the axis of the parabola. Lecture Note 10.1 Conics and Calculus Ellipse An ellipse is the set of points in a plane the sum of whose distances from two fixed points 퐹1 and 퐹2 is a constant. These two fixed points are called the foci (plural of focus). The midpoint of the line segment connecting the foci is called the center. The line passing through two foci is called the major axis. The perpendicular line to the major axis passing through the midpoint of two foci is called the minor axis. The two intersection points of the major axis and the ellipse are called vertices.

minor axis (0, 푏) 푃(푥, 푦) Standard Equation vertex vertex Let the sum of the distances from a point on the ellipse to the foci be 2푎. Then (−푎, 0) (0,0) (푎, 0) 퐹1 퐹2 푥2 푦2 (−푐, 0) (푐, 0) major axis + = 1 푎2 푏2 푐2 = 푎2 − 푏2 < 푎2. (0, −푏) where Lecture Note 10.1 Conics and Calculus

minor axis (0, 푏) Standard Equation 푃(푥, 푦) Let the sum of the distances from vertex vertex a point on the ellipse to the foci be 2푎. Then 2 2 (−푎, 0) (0,0) (푎, 0) 푥 푦 퐹1 퐹2 + = 1 (−푐, 0) (푐, 0) major axis 푎2 푏2 where 푏2 = 푎2 − 푐2 < 푎2. (0, −푏) major axis (0, 푎) vertex Standard Equation (0, 푐) Let the sum of the distances from 퐹 1 푃(푥, 푦) a point on the ellipse to the foci (−푏, 0) be 2푎. Then (푏, 0) 푥2 푦2 (0,0) + = 1 minor axis 푏2 푎2 퐹 푏2 = 푎2 − 푐2 < 푎2. (0, −2푐) where vertex (0, −푎) Lecture Note 10.1 Conics and Calculus

Theorem 10.3 Standard Equation of an Ellipse with Translations The standard form of the equation of a parabola with center (ℎ, 푘) and major and minor axes of lengths 2푎 and 2푏, 푎 > 푏, is 푥 − ℎ 2 푦 − 푘 2 + = 1 Major axis is horizontal 푎2 푏2 or 푥 − ℎ 2 푦 − 푘 2 + = 1 Major axis is vertical 푏2 푎2 The foci lie on the major axis, 푐 units from the center, with 푐2 = 푎2 − 푏2 The ration 푒 = 푐/푎 is called the eccentricity of an ellipse.

Example 3. Find the center, vertices, and foci of the ellipse 4푥2 + 푦2 − 8푥 + 4푦 − 8 = 0. Lecture Note 10.1 Conics and Calculus Applications of Ellipse Movements of Planets in Our Solar System The planets in our solar system move in elliptical orbits, with the Sun at a focus. Earth satellites also travel in elliptic orbits, with Earth at a focus. Lecture Note 10.1 Conics and Calculus Applications of Ellipse Theorem 10.4 Reflection Property of an Ellipse Let 푃 be a point on an ellipse. The tangent line to the ellipse at point 푃 makes equal angles with the lines through 푃 and the foci. Lecture Note 10.1 Conics and Calculus Example Find the area of the ellipse 푥2 푦2 + = 1. 푎2 푏2

푥2 푦2 Example 5. Show that the circumference of the ellipse + = 1 is 푎2 푏2 휋 2 푐 4푎 1 − 푒2 sin2 휃 푑휃, where 푒 = . 0 푎 Elliptic Integral

Example 6. Use the elliptic integral in Example 5 and Simpson’s Rule with 푛 = 4 to approximate the circumference of the ellipse 푥2 푦2 + = 1 25 16 Lecture Note 10.1 Conics and Calculus Hyperbolas An hyperbola is the set of points in a plane the difference of whose distances from two fixed points 퐹1 and 퐹2 is a constant. These two fixed points are called the foci. The peaks are called the vertices of the hyperbola. 푏 푏 An hyperbola has two slant asymptotes: 푦 = 푥 and 푦 = − 푥 푎 푎

푏 푦 = 푥 푎 푃(푥, 푦) Standard Equation Let the difference of the distances from a point on the ellipse to the ±2푎 (0,0) foci be . Then 2 2 (−푐, 0)퐹1 퐹2(푐, 0) 푥 푦 − = 1 푎2 푏2 where 푐2 = 푎2 + 푏2. vertices 푏 (−푎, 0) and (푎, 0) 푦 = − 푥 푎 Lecture Note 10.1 Conics and Calculus 푏 푦 = 푥 푎 Standard Equation 푃(푥, 푦) Let the difference of the distances from a point on the ellipse to the foci be ±2푎. Then (0,0) 2 2 (−푐, 0)퐹 퐹 (푐, 0) 푥 푦 1 2 − = 1 푎2 푏2 where 푐2 = 푎2 + 푏2. vertices 푦 = ± 푏/푎 푥. 푏 Asymptotes: (−푎, 0) and (푎, 0) 푦 = − 푥 푎

푎 푃(푥, 푦) 푦 = 푥 푏 Standard Equation (0, 푐) Let the difference of the distances 퐹2 from a point on the ellipse to the vertices (0, 푎) and (0, −푎) foci be ±2푎. Then (0,0) 푦2 푥2 − = 1 푎2 푏2 2 2 2 퐹1 where 푐 = 푎 + 푏 . (0, −푐) Asymptotes: 푦 = ± 푎/푏 푥. 푎 푦 = − 푥 푏 Lecture Note 10.1 Conics and Calculus

Theorem 10.5 Standard Equation of a Hyperbola The standard form of the equation of a parabola with center at (ℎ, 푘) is 푥 − ℎ 2 푦 − 푘 2 − = 1 푇푟푎푛푠푣푒푟푠푒 푎푥𝑖푠 𝑖푠 ℎ표푟𝑖푧표푛푡푎푙 푎2 푏2 or 푦 − 푘 2 푥 − ℎ 2 − = 1 푇푟푎푛푠푣푒푟푠푒 푎푥𝑖푠 𝑖푠 푣푒푟푡𝑖푐푎푙 푎2 푏2 The vertices are 푎 units from the center, and the foci are 푐 units from the center, where 푐2 = 푎2 + 푏2. The eccentricity 푒 of a hyperbola is given by the ratio 푒 = 푐/푎 (> 1).

Theorem 10.6 Asymptotes of a Hyperbola For a horizontal transverse axis, the equation of the asymptotes are 푏 푏 푦 = 푘 + 푥 − ℎ 푎푛푑 푦 = 푘 − 푥 − ℎ . 푎 푎 For a vertical transverse axis, the equation of the asymptotes are 푎 푎 푦 = 푘 + 푥 − ℎ 푎푛푑 푦 = 푘 − 푥 − ℎ . 푏 푏 Lecture Note 10.1 Conics and Calculus

Graphing Hyperbola 1. Locate the vertices. 2. 2. Use DASHED lines to draw the rectangle centered at the origin with sides parallel to the axes, crossing one axis at ±푎 and the other at ±푏. 3. Use dashed lines to draw the diagonals of this rectangle and extend them to obtain the asymptotes. 4. Draw the two branches of the hyperbola by starting at each vertex and approaching the asymptotes.

Example 7. Sketch the graph of the hyperbola: 4푥2 − 푦2 = 16. What are the equations of the asymptotes?

End of Chapter 10.1 Lecture Note 10.2 Plane Curves and Parametric Equations To this point we’ve looked almost exclusively at functions in the form 푦 = 푓(푥) or 푥 = ℎ(푦) and almost all of the formulas that we’ve developed require that functions be in one of these two forms. The problem is that not all curves or equations that we’d like to look at fall easily into this form.

Take, for example, a circle. It is easy enough to write down the equation of a circle centered at the origin with radius 푟. 푥2 + 푦2 = 푟2 However, we will never be able to write the equation of a circle down as a single equation in either of the forms above. Unfortunately we usually are working on the whole circle, or simple can’t say that we’re going to be working only on one portion of it.

There are also many curves out there that we can’t even write down as a single equation in terms of only 푥 and 푦. Lecture Note 10.2 Plane Curves and Parametric Equations Imagine that a particle moves along the curve 퐶 shown in the figure. It is impossible to describe 퐶 by an equation of the form 푦 = 푓(푥) because 퐶 fails the Vertical Line Test. But we would give its location(푥- and 푦- coordinates) in terms of time: 푥 = 푥(푡) and 푦 = 푦(푡). Such a pair of equations is often a convenient way of describing a curve and gives rise to the following definition.

Definition: If 푓 and 푔 are continuous functions of 푡 (called a parameter) on an interval 퐼, then the equations 푥 = 푓 푡 and 푦 = 푔(푡) are parametric equations . The set of points 푥, 푦 = (푓 푡 , 푔 푡 ) obtained as 푡 varies over the interval 퐼 is the graph of the parametric equations. Taken together, the parametric equations and the graph are a plane curve, (or parametric curve) denoted by 퐶. When the parameter 푡 is in limited range 푎 ≤ 푡 ≤ 푏, the point (푥 푎 , 푦 푎 ) is called the initial point and (푥 푏 , 푦 푏 ) the terminal point. The direction in which the graph of a pair of parametric equations is traced as the parameter increases is called the direction of increasing parameter or sometimes the orientation is imposed on the curve by the equation. Lecture Note 10.2 Plane Curves and Parametric Equations

Sketching the Plane Curve of parametric equations Method 1) Find sample points as 푡 varies, then plot them and connect with curves. Method 2) Find a rectangular equation by eliminating the parameter, then sketch the equation (eliminating the parameter). Domain should be adjusted.

Parametric Solve for 푡 in Substitute into Rectangular equations one equation second equation equation

Use an identity formula to get a rectangular equation Lecture Note 10.2 Plane Curves and Parametric Equations Example 1 Sketch each curve described by the parametric equations 푡 푎 푥 = 푓 푡 = 푡2 − 4 and 푦 = 푔 푡 = , where − 2 ≤ 푡 ≤ 3. 2 푡 푏 푥 = 푓 푡 = 푡2 − 4 and 푦 = 푔 푡 = , where − 1 ≤ 푡 ≤ 3/2. 2 푡 푐 푥 = 푓 푡 = 푡2 − 4 and 푦 = 푔 푡 = − , where − 3 ≤ 푡 ≤ 2. 2 Lecture Note 10.2 Plane Curves and Parametric Equations

Example 2 Sketch the curve described by the parametric equations 1 푡 푥 = and 푦 = , where 푡 > −1 푡 + 1 푡 + 1 by eliminating the parameter and adjusting the domain of the resulting rectangular equation. Include the orientation of the curve. Lecture Note 10.2 Plane Curves and Parametric Equations

Example 3 Sketch the curve described by the parametric equations 푥 = 3 cos 휃 and 푦 = 4 sin 휃 , 0 ≤ 휃 ≤ 2휋 by eliminating the parameter and finding the corresponding rectangular equations. Include the orientation of the curve.

The curve of the parametric equations 푥 = ℎ + 푎 cos 휃 푎푛푑 푦 = 푘 + 푏 sin 휃 , 0 ≤ 휃 ≤ 2휋 is the ellipse (traced counterclockwise) given by 푥 − ℎ 2 푦 − 푘 2 + = 1. 푎2 푏2

The curve of the parametric equations 푥 = ℎ + 푎 sin 휃 푎푛푑 푦 = 푘 + 푏 cos 휃 , 0 ≤ 휃 ≤ 2휋 is also the ellipse (traced clockwise) given by 푥 − ℎ 2 푦 − 푘 2 + = 1. 푎2 푏2 Lecture Note 10.2 Plane Curves and Parametric Equations

Finding Parametric Equations We will now investigate the reverse problem. How can we determine a set of parametric equations for a given graph or a given physical description?

Example 4 Find a set of parametric equations that represents the graph of 푦 = 1 − 푥2, using each of the following parameters. 푑푦 a 푡 = 푥 b The slope 푚 = at the point (푥, 푦) 푑푥

Example (Exercise #40, 45, 46) Find a set of parametric equations that represents the unit circle centered at (0, 0).

Example (Exercise #39, 43, 44)

Find a set of parametric equations of the line segment connecting (푥1, 푦1) and (푥2, 푦2). Lecture Note 10.2 Plane Curves and Parametric Equations

Example 5: Cycloid The curve traced out by a point 푃 on the circumference of a circle as the circle rolls along a straight line is called a cycloid.

Parametric Equations for the cycloid 푥 = 푎 휃 − sin 휃 푦 = 푎(1 − cos 휃) , 휃 ∈ 푅 Lecture Note 10.2 Plane Curves and Parametric Equations

Parametric Equations for the cycloid 푥 = 푎 휃 − sin 휃 푦 = 푎(1 − cos 휃) , 휃 ∈ 푅 Brachistochrone problem: Find the curve along which a particle will slide in the shortest time (under the influence of gravity) from a point 퐴 to a lower point 퐵 not directly beneath 퐴. John Bernoulli showed that the particle will take the least time if the curve is part of an inverted arch of a cycloid. Tautochrone problem: No matter where a particle is placed on an inverted cycloid, it takes the same time to slide to the bottom. Huygens proposed that pendulum clocks should swing in cycloidal arcs because then the pendulum would take the same time to make a complete oscillation whether it swings through a wide or a small arc. Lecture Note 10.2 Plane Curves and Parametric Equations

Definition of a Smooth Curve A curve 퐶 represented by parametric equations 푥 = 푓(푡) and 푦 = 푔(푡) on an interval 퐼 is called smooth when 푓′ and 푔′ are continuous on 퐼 and not simultaneously 0, except possibly at the endpoints of 퐼. The curve 퐶 is called piecewise smooth when it is smooth on each subinterval of some partition of 퐼.

Example Find all points at which the curve given by 푥 = 푡2 and 푦 = 푡4 − 푡2 is not smooth.

Example Find all points at which the curve given by 푥 = 2 (휃 − sin 휃) and 푦 = 2(1 − cos 휃) is not smooth.

End of Chapter 10.2 Lecture Note 10.3 Parametric Equations and Calculus Theorem 10.7 Parametric Form of the Derivative: If a smooth curve 퐶 is given by the equations 푥 = 푓(푡) and 푦 = 푔 푡 then the slope of 퐶 at (푥, 푦) is 푑푦 푑푦 푑푥 = 푑푡 , ≠ 0. 푑푥 푑푥 푑푡 푑푡 The higher-order derivatives are 푑 푑푦 푑2푦 푑 푑푦 푑푡 푑푥 = = 푑푥2 푑푥 푑푥 푑푥 푑푡 푑 푑2푦 푑3푦 푑 푑2푦 푑푡 푑푥2 = = 푑푥3 푑푥 푑푥2 푑푥 푑푡

Example 1. Find 푑푦/푑푥 and 푑2푦/푑푥2 for the curve given by 푥 = sin 푡 and 푦 = cos 푡. Lecture Note 10.3 Parametric Equations and Calculus

Example 2. Find the curve given by 1 푥 = 푡 and 푦 = 푡2 − 4 , 푡 ≥ 0 4 Find the slope and concavity at the point 2, 3 . Lecture Note 10.3 Parametric Equations and Calculus Example 3. The prolate cycloid given by 푥 = 2푡 − 휋 sin 푡 and 푦 = 2 − 휋 cos 푡 crosses itself at the point (2, 0), as shown in the figure. Find the equations of both tangent lines at this point. Lecture Note 10.3 Parametric Equations and Calculus Theorem 10. 8 Arc Length:

We know that the length of a curve 퐶 given in the form 푦 = 퐹 푥 , 푥0 ≤ 푥 ≤ 푥1 is 2 푥1 푑푦 퐿 = 1 + 푑푥 . 푑푥 푥0 Suppose that 퐶 can also be described by the parametric equations 푥 = 푓(푡), and 푦 = 푔(푡), 푎 ≤ 푡 ≤ 푏, where 푥′ and 푔′ are continuous on [푎, 푏] and 퐶 is traversed exactly once as 푡 increases from 훼 to 훽, then the length of 퐶 is 푏 푑푥 2 푑푦 2 푏 퐿 = + 푑푡 = 푓′ 푡 2 + 푔′ 푡 2 푑푡 푎 푑푡 푑푡 푎 Lecture Note 10.3 Parametric Equations and Calculus

Theorem 10. 8 Arc Length:

푏 푑푥 2 푑푦 2 푏 퐿 = + 푑푡 = 푓′ 푡 2 + 푔′ 푡 2 푑푡 푎 푑푡 푑푡 푎

Example 4. A circle of radius 1 rolls around the circumference of a larger circle of radius 4, as shown in the figure. The curve traced by a point on the circumference of the smaller circle (calls an epicycloid) is given by 푥 = 5 cos 푡 − cos 5푡 and 푦 = 5 sin 푡 − sin 5푡 . Find the distance traveled by the point in one complete trip about the larger circle. Lecture Note 10.3 Parametric Equations and Calculus

Theorem 10.9 Surface Area of Revolution: We know that the surface area of a solid obtained by rotating the curve 퐶 given by 푦 = 푓 푥 , 푥0 ≤ 푥 ≤ 푥1 about 푥-axis is 2 푥1 푑푦 푆퐴 = 2휋 푦 1 + 푑푥 . 푑푥 푥0 Suppose that 퐶 can also be described by the parametric equations 푥 = 푓(푡), and 푦 = 푔(푡), 푎 ≤ 푡 ≤ 푏, where 푥′ and 푦′ are continuous on [푎, 푏]. If the curve is rotated about the 푥-axis, then the area of the resulting surface is given by 2 2 푏 푑푥 푑푦 푆퐴 = 2휋 푔(푡) + 푑푡 Revolution about the 푥 axis: 푔 푡 ≥ 0 푎 푑푡 푑푡 2 2 푏 푑푥 푑푦 푆퐴 = 2휋 푓(푡) + 푑푡 Revolution about the 푦 axis: 푓 푡 ≥ 0 푎 푑푡 푑푡 Lecture Note 10.3 Parametric Equations and Calculus Theorem 10.9 Surface Area of Revolution:

2 2 푏 푑푥 푑푦 푆퐴 = 2휋 푔(푡) + 푑푡 Revolution about the 푥 axis: 푔 푡 ≥ 0 푎 푑푡 푑푡 2 2 푏 푑푥 푑푦 푆퐴 = 2휋 푓(푡) + 푑푡 Revolution about the 푦 axis: 푓 푡 ≥ 0 푎 푑푡 푑푡

Example 5. Let 퐶 be the arc of the circle 푥2 + 푦2 = 9 from 3 3 3 (3, 0) to , as shown in the figure. Find the area of the 2 2 surface formed by revolving 퐶 about the 푥-axis.

End of Chapter 10.2