1. 2003 ICTM Contest Division AA Orals Topic: Conics Micah Fogel Illinois Math and Science Academy

2. The Source There is no official source this year. Source was to be Schaum’s Outline: Analytic Geometry Out of print!

3. Topics Locus definitions of conics Focus/directrix definitions of conics Rectangular equations Parametric equations Translation and rotation of axes

4. Topics (State level) Polar forms for conics

5. Additional Resources Mailing list: fogel@imsa.edu Webstie: http://www.imsa.edu/~fogel ICTM Website: www.viebach.net/ictm/admin/c2003r.htm Books –Analytic geometry –Calculus –Last few orals sources

6. Study Ideas Study each aspect by itself (i.e., rectangular equations one day, proofs of reflection properties another day, etc.) Steal lots of basic exercises so that the algebra becomes automatic Try to find different explanations of the relationships between the various properties of conics

7. Cones and Conic Sections

8. Slicing a Cone Intersect a double-cone with a plane. If the plane goes through the vertex of the cones, the intersection is either a point, a line, or two crossing lines.

9. Slicing a Cone These three cases are degenerate and are not really very interesting The interesting stuff happens when the plane does not go through the vertex Then there are three cases –Plane has less slant than cone –Plane has more slant than cone –Plane is parallel to slant of cone

10. Ellipse In the first case, there are two spheres, called Dandelin (Belgian, 1794–1847) spheres, which are tangent to both cone and plane The points where these spheres touch the plane will be called foci (sing.: focus)

11. Ellipse The intersection of the plane with the cone is called an ellipse Pick a point on the ellipse and draw the line through it and the vertex of the cone The distance between the two dashed circles on this line is constant

12. Ellipse But the distances to the foci are the same as the distances along this line, since they are tangents to the same spheres Thus, for any point on an ellipse, the sum of the distances to the foci is a constant

13. Ellipse Concentrate on one of the Dandelin spheres. Consider the plane through the circle where it is tangent to the cone Call the line where this plane intersects the cutting plane a directrix

14. Ellipse From a point P on the ellipse, travel parallel to the axis of the cone to the new plane, call this point Q. Travel along the cone until the intersection of the cone and sphere is reached. Call this point A. Choose D so that QD  directrix Q P A F D

15. Ellipse Note that in triangle PQD, we have PQ = PD cos , where  is the angle between the cutting plane and the cone axis P D Q  Q P D

16. Ellipse Also note that angle  QPA has the same measure as the angle between the axis and slant of the cone. So PQ = PA cos  Also note PF = PA, since both are tangents to a sphere from P Q P A F D

17. Ellipse Putting these relationships together gives PD cos  = PF cos , or that Thus, since  and  are constants, the ratio of the distances from a point on the ellipse to a focus and to the corresponding directrix is a constant, called the eccentricity

18. Ellipse—Summary An ellipse is the intersection of a cone and a plane with less slant than the cone The ellipse has two foci, which are the intersection of the slant plane and the Dandelin spheres The sum of the distances from the foci to any point on the ellipse is a constant

19. Ellipse—Summary The slant plane meets the plane through the circle where a Dandelin sphere intersects the cone in a line called the directrix The distance from any point on the ellipse to a focus is a constant—the eccentricity— times the distance from that point to the directrix Since the plane has less slant than the cone, the eccentricity is less than one

20. Hyperbola If the slant of the plane is more than the slant of the cone, then the plane intersects both nappes of the cone This time, both Dandelin spheres are on the same side of the plane Thus, everything works the same, except that we have to subtract distances to foci to get a constant, and the eccentricity will be larger than one

22. Hyperbola—Summary A hyperbola is the intersection of a cone and a plane with more slant than the cone The hyperbola has two foci, which are the intersection of the slant plane and the Dandelin spheres The difference of the distances from the foci to any point on the hyperbola is a constant

23. Hyperbola—Summary The slant plane meets the plane through the circle where a Dandelin sphere intersects the cone in a line called the directrix The distance from any point on the hyperbola to a focus is a constant—the eccentricity— times the distance from that point to the directrix Since the plane has more slant than the cone, the eccentricity is greater than one

24. Parabola When the slant plane has the same slant as the cone itself, the intersection is a single curve called a parabola This time, there can only be one Dandelin sphere So a parabola has only one focus

26. Parabola We can repeat all the same arguments and calculations to find: –A parabola is the intersection of a cone with a plane of the same slant –Because slants are equal, there is only one Dandelin sphere, which meets the plane in the focus –Because slants are equal, eccentricity is one, so parabola is the set of points equidistant from a line (the directrix) and a point (the focus)

27. Circle A special case of a plane intersecting a cone is if the plane is perpendicular to the axis of the cone, resulting in a circle If we repeat all the above constructions, there are two Dandelin spheres, which are both tangent to the cutting plane—circles have only one “focus,” the center The relevant planes are parallel, so circles have no directrices. Eccentricity is zero

28. Reflection Principle Given two points on the same side of a line, the shortest path from one to the other which touches the line meets the line at equal angles coming and going This can be seen by reflecting one point across the line, and knowing the shortest distance between the other point and the reflection is a straight line

29. Reflection Principle P1P1 P2P2 P2’P2’

30. Reflection Propety of Ellipses Apply this to the foci and a tangent line to an ellipse F 1 QF 2 > F 1 PF 2 since F 1 P’F 2 = F 1 PF 2, so P is the point at which incident angles are equal F1F1 F2F2 P Q P’P’

31. Reflection Property of Parabolas If we move the focus F 2 more and more to the right, the reflected ray becomes more and more horizontal If we let F 2 “go to infinity” the ellipse turns into a parabola So a ray from the focus of a parabola will be reflected parallel to the axis of the parabola

32. Reflection Property of Hyperbolas If you keep moving F 2 to the right, “past” infinity, it reappears on the left. The ellipse turns inside-out, and becomes a hyperbola So the corresponding reflection property for hyperbolas is that a ray from one focus will be reflected off the hyperbola directly away from the other focus (Of course, these aren’t rigorous proofs for parabola and hyperbola!)

33. Reflection Property of Hyperbolas

34. Cartesian Equations for Conics We can use the distance formula and the focal distance properties of conics to find equations for their graphs in the Cartesian plane For instance, a circle with radius r and center (h, k) is simply Squaring, we obtain the standard equation (x - h) 2 + (y - k) 2 = r 2

35. Cartesian Equation for Ellipse If we put the foci at (  c, 0) and make the total distance of a point to the two foci 2a the distance formula gives us Move one radical to the other side, square, and simplify to obtain

36. Cartesian Equation for Ellipse Isolate the radical and square again: a 2 (x 2 + 2cx + c 2 + y 2 ) = a 4 + 2a 2 cx + c 2 x 2 This simplifies to (with b 2 = a 2 - c 2 ) Eccentricity e = c/a

37. Cartesian Equation for Hyperbola The derivation for a hyperbola works exactly the same, except for the sign difference, yielding (Foci are at (  c, 0), difference between focal distances is 2a, and c 2 = a 2 + b 2, eccentricity e = c/a)

38. Cartesian Equation for Parabola If we put the focus at (0, c) and the directrix as the line y = -c, we get the equation Squaring and simplifying, we obtain x 2 = 4cy Naturally, we can translate any of these equations to have center (h, k) by replaceing x with (x - h) and y with (y - k)

39. Info from the Equations We should be able to go from descriptions (“ellipse with its foci at (2, 5) and (2, 1) and eccentricity 0.4”) to Cartesian equations (here, center is (2, 3) so c = 2, c/a = 0.4 and a = 5, and therefore b 2 = 5 2 - 2 2 = 21, giving the equation (x - 2) 2 /21 +(y - 3) 2 /25 = 1) We should be able to go backward, finding center, foci, directrices, eccentricity, etc. from the Cartesian equation

40. General Quadratic Equation The general quadratic equation looks like Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 With no cross-product term, we could complete the square and reduce to one of our standard equations So we get rid of the cross-product term If we replace x with x’cos  - y’sin  and replace y with x’sin  + y’cos  we get….

41. General Quadratic Equations A’x’ 2 + B’x’y’ + C’y’ 2 + D’x’ + E’y’ + F’ = 0 We can obtain formulas for A’, B’, etc. The only important one is B’ = Bcos(2  ) +(C - A)sin(2  ) So if we set tan(2  ) = B/(A - C), B’ will equal zero

42. Rotation of Axes This process is called rotation of axes The new x’ and y’ axes are at angle  to the original axes We can rotate the axes to eliminate the cross-product term, then complete the squares to reduce the equation to standard form

43. Example Tell about 2x 2 + 24xy + 9y 2 +5x - 10y - 30 = 0 First, if tan(2  ) = -24/7, cos(  ) = 3/5 and sin(  ) = 4/5 So take x = 3x’/5 - 4y’/5, y = 4x’/5 + 3y’/5 Substitute to obtain 18x’ 2 - 7y’ 2 - 5x’ - 10y’ - 30 = 0

44. Example 18(x’ 2 - 5/18 x’) - 7(y’ 2 - 10/7y’) - 30 = 0 18(x’ 2 - 5/18 x’ + 25/1296) - 7(y’ 2 - 10/7 y’ +25/49) - 30 - 25/72 + 25/7 = 0 18(x’ - 5/36) 2 - 7(y’ - 5/7) 2 = 16745/504

45. Discriminant The combination B 2 - 4AC is left unchanged by any rotation of axes In particular, with the correct rotation, B = 0 AC then tells whether you have an ellipse (both same sign), hyperbola (opposite sign) or parabola (one is zero) B 2 - 4AC is called the discriminant. If negative, we have an ellipse. Positive gives a hyperbola. Zero gives a parabola.

46. Degenerate Conics If the discriminant is positive, the hyperbola may degenerate into a pair of crossing lines If the discriminant is negative, the ellipse might be a circle, a single point, or an empty graph If the discriminant is zero, the parabola may degenerate into a straight line, a pair of parallel lines, or an empty graph

47. Asymptotes Hyperbolas have asymptotes For the asymptotes are found by setting the right-hand side to 0 Then we get the lines y - k =  (b/a)(x - h)

48. Parametric Equations Using the trig relations cos 2  + sin 2  = 1 and sec 2  - tan 2  = 1 we can transform any conic into a parametric equation Let x = h + a cos  and y = k+ b sin . Then (x - h)/a = cos , (y - k)/b = sin , so we have the ellipse

49. Parametric Equations Similarly, a general hyperbola might be given by x = h + tan , y = k + sec  (for hyperbolas that open up and down) or by x = h + sec , y = k + tan  (for hyperbolas that open to the sides) Parabolas can have y expressed as a function in x, or vice-versa, which are easy to turn into parametrics—let x (or y) be the parameter!

50. Polar Coordinates Using the directrix-focus properties of conics, it is easy to find equations for conics in polar coordinates. Recall x = r cos  and y = r sin  Put a focus at the origin and make the corresponding directrix the line x = p Recall that all conics have the distance from a point to the focus e (eccentricity) times the distance from the point to the directrix

51. Polar Coordinates So if (x, y) is on a conic we have that r equals e(x - p) Using x = r cos , we get r = e(r cos  - p) or r(1 - e cos  ) = -ep Thus the equation becomes We can rotate this to have different orientations by manipulating 