9.1 Defining the conic sections in terms of their locus definitions. Part 1

What’s a locus definition? Locus definitions describe the location of a set of points using mathematical criteria. Ex. Where are all the points which are equidistant from the endpoints of a segment? The points equidistant from the endpoints of a segment lie on the perpendicular bisector of the segment.

Conic sections are shapes which are based on a cone. If you slice across the cone, parallel to the base, what shape will you get?

What would be the locus definition of a circle? A circle is the set of all points which are a constant distance from a fixed point (the center).

Imagine now that the center of the circle splits into two points that move away from each other. As these points move, the circle deforms into an elliptical shape to accommodate these two “centers.” These points are called focal points or foci.

In a circle, the distance between the center and any point is constant. From each point on an ellipse, there are two distances – one to each focal point. F1 F2

Refer to the diagram on your note sheet Refer to the diagram on your note sheet. Use a ruler to measure the distance from A to the focal points and record in your table. Then mark a point in 4 different places around the ellipse and again measure the distances from those points to the foci and record in your table. Add the distances for each point. What is true about the sum?

F1 F2 An ellipse is the set of points such that the sum of the distances from each point on the ellipse to two fixed points (the foci) is constant.

String Activity

To satisfy the definition of an ellipse, the sum of the distances from each point to the foci must be constant. In this construction, the sum of the distances from the pencil to each taped end of the string remains constant because the string length does not change. Therefore, the pencil traces out an ellipse. 2. The focal points are the taped ends of the string. 3. BF1 and CF2 appear to be equal in length. 4. When the ends of the string are taped to the same spot, your pencil will draw a circle.

5. If the string is extended so there is no slack, then the pencil will draw a line segment from one foci (taped end) to the other. 6. When the ends of the string are closer together, the ellipse will be more circular. The major axis is the longest chord of the ellipse. The minor axis is the shortest chord of the ellipse that passes through the center of the ellipse.

The length of the string is equal to the length of the major axis. 9. Since the length of the string is equal to the length of the major axis then DF1 + DF2 = 9.5. Because of the symmetry of the ellipse, DF1 and DF2 are the same length and thus both must be 4.75 cm long.

10. a). the minor axis is bisected by the major axis and DO = EO = 3 10. a) the minor axis is bisected by the major axis and DO = EO = 3.6 cm. b) From #9, we know that from top of the ellipse to the focal points is 4.75 cm. Triangle DOF2 is a right triangle so we can use the Pythagorean Theorem to find OF2 which will be the same as OF1.

11.

Ex. If the major axis of an ellipse is 12 cm and the minor axis is 8 cm, how far apart are the focal points? Distance between foci = Focal Length = 8.94

Ex. If the focal length of an ellipse is 20 cm and the minor axis is 10 cm, what is the length of the major axis? Length of major axis = 22.36

For your fun and enjoyment…