§ 10.2 The Ellipse

The Conic Sections Conic sections are the curves that result from the intersection of a right circular cone and a plane. There are four conic sections: the circle, the ellipse, the parabola and the hyperbola. You can see your text on page 742 to see how these curves are formed from that intersection of a plane and a cone. The conics occur naturally throughout the universe. The Ancient Greeks began studying these curves more than 2000 years ago, simply because studying them was exciting, interesting, and challenging. The Ancient Greeks could not have imagined the applications of these curves in our world today. The conics enable the Hubble Space Telescope to gather distant rays of light and focus them into spectacular images of our evolving universe. They provide doctors with a procedure for dissolving kidney stones painless without invasive surgery. There are even applications of conics that move beyond our planet. Ever studied Haley’s Comet? In this section, we study the symmetric oval-shaped curve known as the ellipse. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 10.2

Drawing an Ellipse Drawing an ellipse: 1. Place straight pins at two fixed points, each of which is called a focus (foci is the plural) 2. Take the ends of a fixed length of string and fasten the ends of the string to the pins 3. Draw the string taut with a pencil 4. Trace a path with the pencil The oval shaped curve which you have drawn is called an ellipse. This procedure for drawing an ellipse illustrates its definition: An ellipse is the set of all points the sum of whose distances from two fixed points in the plane is constant. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 10.2

Definition of an Ellipse Equation of an Ellipse Definition of an Ellipse An ellipse is the set of all points, P, in a plane the sum of whose distances from two fixed points, , is constant. These two fixed points are called the foci (plural of focus). The midpoint of the segment connecting the foci is the center of the ellipse. Blitzer, Intermediate Algebra, 5e – Slide #4 Section 10.2

Standard Forms of the Equations of an Ellipse Equation of an Ellipse Standard Forms of the Equations of an Ellipse The standard form of the equation of an ellipse with center at the origin, and major and minor axes of lengths 2a and 2b (where a and b are positive, and ) is The figures below illustrate that the vertices are on the major axis, a units from the center. The foci are on the major axis, c units from the center. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 10.2

Standard Forms of the Equations of an Ellipse Equation of an Ellipse CONTINUED Standard Forms of the Equations of an Ellipse (0,a) (0,b) (0,c) (0,0) (0,0) (-b,0) (b,0) (-a,0) (-c,0) (c,0) (a,0) (0,-b) (0,-c) (0,-a) Major axis is horizontal with length 2a. Major axis is vertical with length 2a. Blitzer, Intermediate Algebra, 5e – Slide #6 Section 10.2

Equation of an Ellipse Graph the ellipse: EXAMPLE Graph the ellipse: SOLUTION We begin by expressing the equation in standard form. Because we want 1 on the right side, we divide both sides by 100. This is the larger of the two denominators. This is the smaller of the two denominators. Blitzer, Intermediate Algebra, 5e – Slide #7 Section 10.2

Equation of an Ellipse CONTINUED The equation is the standard form of an ellipse’s equation with Because the denominator of the is greater than the denominator of the , the major axis is horizontal. Based on the standard form of the equation, we know that the vertices are (a, 0) and (-a, 0). Because , a = 5. Thus, the vertices are (5, 0) and (-5, 0). Now let us find the endpoints of the vertical minor axis. According to the standard form of the equation, these endpoints are (0, b) and (0, -b). Because , b = 2. Thus the endpoints of the minor axis are (0, 2) and (0, -2). Using the four endpoints, we sketch the ellipse below. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 10.2

Equation of an Ellipse CONTINUED (0,2) Vertex (-5,0) Vertex (5,0) (0,-2) Blitzer, Intermediate Algebra, 5e – Slide #9 Section 10.2

Standard Forms of Equations of Ellipses Centered at (h, k) Equation of an Ellipse Standard Forms of Equations of Ellipses Centered at (h, k) Equation Center Major Axis Vertices (h, k) Parallel to x-axis, horizontal (h - a, k) (h + a, k) Graph y Vertex (h + a, k) Major axis (h, k) Vertex (h - a, k) x Blitzer, Intermediate Algebra, 5e – Slide #10 Section 10.2

Standard Forms of Equations of Ellipses Centered at (h, k) Equation of an Ellipse CONTINUED Standard Forms of Equations of Ellipses Centered at (h, k) Equation Center Major Axis Vertices (h, k) Parallel to y-axis, vertical (h, k - a) (h, k + a) Graph y Vertex (h, k + a) Major axis (h, k) Vertex (h, k - a) x Blitzer, Intermediate Algebra, 5e – Slide #11 Section 10.2

Equation of an Ellipse Graph the ellipse: EXAMPLE Graph the ellipse: SOLUTION To graph the ellipse, we need to know its center, (h, k). In the standard forms of equations centered at (h, k), h is the number subtracted from x and k is the number subtracted from y. This is with k = 2. This is with h = -3. We see that h = -3 and k = 2. Thus, the center of the ellipse, (h, k), is (-3, 2). We can graph the ellipse by locating endpoints on the major and minor axes. To do this, we must identify and Blitzer, Intermediate Algebra, 5e – Slide #12 Section 10.2

Equation of an Ellipse CONTINUED The larger number is under the expression involving x. This means that the major axis is horizontal and parallel to the x-axis. We can sketch the ellipse by locating endpoints on the major and minor axes. Endpoints of the major axis (the vertices) are 3 units to the right and left of the center. Endpoints of the minor axis are 1 unit up and down from the center. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 10.2

Equation of an Ellipse For a Horizontal Major Axis with Center (-3, 2) CONTINUED We categorize the observations in the voice balloons as follows: For a Horizontal Major Axis with Center (-3, 2) Vertices Endpoints of Minor Axis (-3 + 3, 2) = (0, 2) (-3, 2 + 1) = (-3, 3) (-3 - 3, 2) = (-6, 2) (-3, 2 - 1) = (-3, 1) Using the center and these four points, we can sketch the ellipse shown as follows. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 10.2

Equation of an Ellipse CONTINUED (-3,3) (-6,2) (-3,2) (0,2) (-3,1) Blitzer, Intermediate Algebra, 5e – Slide #15 Section 10.2

Equation of an Ellipse EXAMPLE A semielliptic archway has a height of 20 feet and a width of 50 feet as shown in the figure below. Can a truck 14 feet high and 10 feet wide drive under the archway without going into the other lane? SOLUTION Because the right side of the truck is 10 feet from the center of the archway, we must find the height of the archway 10 feet Blitzer, Intermediate Algebra, 5e – Slide #16 Section 10.2

Equation of an Ellipse CONTINUED from the center. If that height is 14 feet or less, the truck will not clear the opening. In the figure below, we’ve constructed a coordinate system with the x-axis on the ground and the origin at the center of the archway. Also shown is the truck, whose height is 14 feet. (0,20) x (25,0) (-25,0) Blitzer, Intermediate Algebra, 5e – Slide #17 Section 10.2

Equation of an Ellipse CONTINUED Using the equation , we can express the equation of the archway as As shown in the figure, the right side edge of the truck corresponds to x = 10. We find the height of the archway 10 feet from the center by substituting 10 for x and solving for y. Substitute 10 for x in Square 10. Blitzer, Intermediate Algebra, 5e – Slide #18 Section 10.2

Equation of an Ellipse CONTINUED Clear fractions by multiplying both sides by the LCD, 10,000. Use the distributive property. Simplify. Subtract 1600 from both sides. Divide both sides by 25. Take only the positive square root. The archway is above the x-axis, so y is nonnegative. Use a calculator. Blitzer, Intermediate Algebra, 5e – Slide #19 Section 10.2

Equation of an Ellipse CONTINUED Thus, the height of the archway 10 week from the center is approximately 18.33 feet. Because the truck’s height is 14 feet, there is enough room for the truck to clear the archway. Whispering galleries… Have you ever been in a whispering gallery? A whispering gallery is an elliptical room with an elliptical, dome-shaped ceiling. People standing at the foci can whisper and hear each other quite clearly, while persons in other locations in the room cannot hear them. Statuary Hall in the U.S. Capitol Building is elliptical. President John Quincy Adams, while a member of the House of Representatives, was aware of this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House Members located near the other focus. Blitzer, Intermediate Algebra, 5e – Slide #20 Section 10.2