Find the center and radius of...

Warm Up x h 2 + y k 2 = r 2 Circle with center h, k and radius r. Find the center and radius of... 2 2 a) ( x 3) y 7 19 2 2 b) x y 6x 4y 12 0

Chapter 6 Analytic Geometry (Conic Sections)

Conic Section A figure formed by the intersection of a plane and a right circular cone

6.2 Equations of Circles x 2 y 2 r 2 Circle with radius r & center (0,0) 2 2 2 x h y k r Find the center and radius of... 2 a) ( x 3) 7 19 (3, 7) y 2 2 2 b) x y 6x 4y 12 0 r 19 Circle with radius r & center (h,k) x 2 + bx + b 2 2 = x + b 2 2 2 2 x 6x y 4y 12 2 2 ( x 6 x 9 ) ( y 4 y 4 ) 12 13 2 2 ( x 3 ) ( y 2 ) 25 (3, 2) r 5

2 2 x 2 + bx + b 2 = x + b 2 Find the radius and center of the circle with the equation x 2 + y 2 2x 6y 9 = 0 x 2 2x + + y 2 6y + = 9 x 2 2x + 1 + y 2 6y + 9 = 9 + 1 + 9 x 1 2 + y 3 2 = 19 center = (1,3) r = 19

6.2 Equations of Circles 2 2 ( x3) ( y 2) 25 Graph the circle by hand and then on a calculator. y Center-Radius Form (3, 2) r 5 3,3 5 x 2, 2 3, 2 8, 2 3, 7

6.2 Equations of Circles 2 2 ( x3) ( y 2) 25 Graph the circle by hand and then on a calculator. 2 2 y 2 25 x 3 Center-Radius Form Solve for y y y 2 25 x 3 2 25 x 3 2 2

6.2 Equations of Circles y x 2 2 25 3

6.2 Equations of Circles TEST PREP: Find the equation of the circle with diameter endpoints of (3,7) and (-2,-4). DIAMETER 2 2 2 3 4 7 146 r 2 2 1 3 midpoint:, 2 2 x y 2 2 1/ 2 3/ 2 73/ 2 2 2 2 x h y k r For any two points P x 1, y 1 and Q(x 2, y 2 ): The length of PQ is x 1 x 2 2 + y 1 y 2 2 The midpoint M of PQ is x 1 +x 2 2, y 1+y 2 2

6.3 Ellipses Objective: To find equations of ellipses and to graph them. Complete the Drawing an Ellipse Activity With Your Group

Drawing an Ellipse Attach the ends of a string to the tacks. With the point of a pencil, hold the string taut. Then, carefully move the pencil around the foci, keeping the string taut at all times.

Drawing an Ellipse The pencil will trace out an ellipse. This is because the sum of the distances from the point of the pencil to the foci will always equal the length of the string, which is constant.

Drawing Ellipses If the string is only slightly longer than the distance between the foci, the ellipse traced out will be elongated in shape.

Drawing Ellipses If the foci are close together relative to the length of the string, the ellipse will be almost circular.

Elliptical Orbits of Planets Gravitational attraction causes the planets to move in elliptical orbits around the sun with the sun at one focus. This remarkable property was first observed by Johannes Kepler. It was later deduced by Isaac Newton from his inverse square law of gravity using calculus.

Reflection Property Ellipses, like parabolas, have an interesting reflection property that leads to a number of practical applications. If a light source is placed at one focus of a reflecting surface with elliptical cross sections, then all the light will be reflected off the surface to the other focus.

Lithotripsy This principle, which works for sound waves as well as for light, is used in lithotripsy a treatment for kidney stones. The patient is placed in a tub of water with elliptical cross sections in such a way that the kidney stone is accurately located at one focus. High-intensity sound waves generated at the other focus are reflected to the stone and destroy it with minimal damage to surrounding tissue. The patient is spared the trauma of surgery and recovers within days instead of weeks.

Whispering Galleries The reflection property is also used in the construction of whispering galleries. Sound coming from one focus bounces off the walls and ceiling of an elliptical room and passes through the other focus. Even quiet whispers spoken at one focus can be heard clearly at the other.

Whispering Galleries Famous whispering galleries include: National Statuary Hall of the US Capitol in Washington, D.C. Mormon Tabernacle in Salt Lake City, Utah

Pioneer Courthouse Square SW 6 th & Morrison Terry Schrunk Plaza SW 4 th & Madison

Minor Axis The Ellipse An ellipse is the locus of all points in a plane such that the sum of the distances from two given points in the plane, the foci, is constant. Focus 1 Focus 2 Major Axis Point PF 1 + PF 2 = constant

Minor Axis The Ellipse An ellipse is the locus of all points in a plane such that the sum of the distances from two given points in the plane, the foci, is constant. Major Axis (a,0) Focus 1 Focus 2 Point PF 1 + PF 2 = 2a

The Standard Form of the Equation of the Ellipse The standard form of an ellipse centered at the origin with the major axis of length 2a along the x-axis and a minor axis of length 2b along the y-axis, is: The standard form of an ellipse centered at the origin with the major axis of length 2a along the y-axis and a minor axis of length 2b along the x-axis, is: (0,c) x 2 a 2 y2 b 2 1 x 2 b 2 y2 a 2 1 (0,-c)

The Pythagorean Property b a F 1 (-c, 0) F 2 (c, 0) c a 2 = b 2 + c 2 b 2 = a 2 - c 2 c 2 = a 2 - b 2

The Standard Forms of the Equation of the Ellipse The standard form of an ellipse centered at any point (h, k) with the major axis of length 2a parallel to the x-axis and a minor axis of length 2b parallel to the y-axis, is: (x h) 2 a 2 (y k)2 b 2 1 Foci: (- c h, k) and ( c h, k) where c 2 = a 2 b 2 (h, k)

The Standard Form of the Equation of the Ellipse The standard form of an ellipse centered at any point (h, k) with the major axis of length 2a parallel to the y-axis and a minor axis of length 2b parallel to the x-axis, is: (x h) 2 b 2 (y k)2 a 2 1 Foci: ( h, - c k) and ( h, c k) where c 2 = a 2 b 2 (h, k)

Finding the Center, Axes, and Foci State the coordinates of the vertices, the coordinates of the foci, and the lengths of the major and minor axes of the ellipse. Then sketch a graph of the ellipse. 2 2 x y 1 9 4 b c a The center of the ellipse is (0, 0). Since the larger number occurs under the x 2, the major axis lies on the x-axis. The length of the major axis is 6. The length of the minor axis is 4. The coordinates of the vertices are (3, 0) and (-3, 0). a = b = 3 2 To find the coordinates of the foci, use the Pythagorean property. c 2 = a 2 - b 2 = 3 2-2 2 = 9-4 = 5 c 5 The coordinates of the foci are: ( 5, 0) and ( 5, 0)

Finding the Equation of the Ellipse With Center at (h, k) Find the equation for the ellipse with the center at (3, 2), passing through the points (8, 2), (-2, 2), (3, -5), and (3, 9). The major axis is parallel to the y-axis and has a length of 14 units, so a = 7. The minor axis is parallel to the x-axis and has a length of 10 units, so b = 5. The center is at (3, 2), so h = 3 and k = 2. (3, 2) (x h) 2 b 2 (x 3) 2 5 2 (y k)2 a 2 1 (y 2)2 7 2 1 (x 3) 2 25 (y 2)2 49 1

Sketch the graph of the ellipse. Include the foci. Since the larger number occurs under the y 2, the major axis is parallel to the y-axis. ( x 1) ( y 5) 16 36 2 2 1 h = k = a = b = 1-5 6 4 c 2 = a 2 - b 2 = 6 2-4 2 = 36-16 = 20 c 20 c 2 5 The center is at (1, -5). The major axis, parallel to the y-axis, has a length of 12 units. The minor axis, parallel to the x-axis, has a length of 8 units. The foci are at: ( 1, 5 2 5) and ( 1, 5 2 5)

Sketching the Graph of an Ellipse (x 1) 2 16 ( y 5)2 36 1 The center is at (1, -5). The major axis, parallel to the y-axis, has a length of 12 units. The minor axis, parallel to the x-axis, has a length of 8 units. The foci are at: ( 1, 5 2 5) and ( 1, 5 2 5) F 1 F 2 (1, 5 2 5 ) c 2 5 c 2 5 (1, -5-2 5)

Graphing an Ellipse Using a Graphing Calculator (x 1) 2 16 ( y 1)2 4 1 y 16 (x 1)2 4 1 (x - 1) 2 + 4(y + 1) 2 = 16 4(y + 1) 2 = 16 - (x - 1) 2 y 1 2 16 (x 1) 2 4 y 1 16 (x 1)2 4 y 16 (x 1)2 4 1 y 16 (x 1)2 4 1

Homework Page 222 #1-19 odd Page 228 #1,3,5,7a,13,15,25

Test Scores Average Median 6 th Period 91.2% 93.3% 7 th Period 89.5% 94.7% 8 th Period 91.1% 92.7%

Eccentricity The eccentricity, e, is a measure of how stretched the ellipse is. If e is close to 1, c is almost equal to a, and the ellipse is elongated in shape. However, if e is close to 0, the ellipse is close to a circle in shape.

Elliptical Orbits of Planets The orbits of the planets have different eccentricities. Most are nearly circular. However, Mercury and the former planet Pluto the innermost and outermost have visibly elliptical orbits.