Essential mathematics: Conics

66 flashcards · Mathematics · by lakesare

1. What's the name of each of these curves?

   (a) A parabola (b) An ellipse (c) A hyperbola

2. Why are these curves (parabola, ellipse, hyperbola) called conics?

   Because we obtain them by slicing double cone with a plane.

3. What's opening angle of a cone?

4. What 2 categories of curves is it possible to produce by slicing the double cone?

   non-degenerate degenerate - when your slicing plane passes through the apex of the double cone.

5. How is parabola defined geometrically? Circle can be defined geometrically as the set of points that are a fixed distance from a particular point.

   We have a line (directrix) a point (focus) Parabola is defined to be a set of points such that PF = Pd.

6. Find the equation of the parabola with focus (5, 0) and directrix x = −5.

   y2 = 20x.

7. What are the coordinates of focus & directrix of a parabola in a standard position?

   focus: F(a, 0) directrix: x = -a

8. What's the equation of a parabola in standard position?

   y2 = 4ax.

9. Write down the equation of the parabola in standard position with the given focus ( 14\frac{1}{4}41​ , 0).

   y2 = x. Find a (here it's 14\frac{1}{4}41​ ). Apply y2 = 4ax formula for parabolas in standard position.

10. Consider the equation y2 = x. (a) Show that this equation represents a parabola in standard position. (b) Sketch the parabola. (c) Write down the coordinates of the focus and the equation of the directrix, and add these features to your sketch.

11. What's parabolic reflector?

    Parabolic reflector is a reflective surface whose shape is formed by rotating part of a parabola through 360° about its axis of symmetry.

12. Explain this picture.

    It can be proved, by using the focus–directrix property of a parabola, If a light source is positioned at the focus of a parabolic reflector, - then the rays of light are reflected in a direction that is parallel to the axis. If rays of light that are parallel to the axis of a parabolic reflector strike its inner surface, then they are reflected to pass through its focus.

13. When light hits the flat surface, it's reflected at the same angle. At what angle does the light reflect when it hits the curved surface?

    Light is reflected at the same angle, - from the tangent at that point. For a curved surface, the two angles are measured from a line that just touches the surface at the point where the ray strikes, as shown

14. Parabolas, ellipses and hyperbolas can be defined by the equation PF = e × Pd. PF - distance from point P to focus FPd - distance from point P to directrix de - eccentricity What values can e take for parabolas ellipses hyperbolas? Each point on parabola is e times as far from the focus as it is from the directrix.

    ellipses - e ∈ (0, 1) E.g. PF = 0.3 × Pdparabolas - e = 1 Only PF = Pdhyperbolas - e ∈ (1, ∞) E.g. Pd = 888 × Pd Ellipse Parabola Hyperbola

15. What's the physical interpretation of the eccentricity (e) for an ellipse?

    If e is 0 - it's a circle. If e is 1 - it's an extreme ellipse (parabola).

16. Parabola VS ellipse in standard position?

17. How would you go about deriving the equation of ellipse and hyperbola?

    It's the same as with the parabola, however more of a headache. It's a good idea to just use the handbook.

18. What's the equation of the ellipse?

19. How many of these do you have to know in order to derive the equation of the ellipse?

    2 facts. If you know any two of the following five pieces of information about an ellipse in standard position, then you can use the facts in the box above to work out any of the other three pieces of information, and the equation of the ellipse

20. What's major&minor axis in an ellipse?

    In any position: major axis - the longer axis minor axis - the shorter axis For an ellipse in standard position, the major axis lies along the x-axis and the minor axis lies along the y-axis.

21. What are the vertices of an ellipse?

    4 points where ellipse crosses axes of symmetry. Since an ellipse is a circle scaled in one direction, it has two axes of symmetry. Since an ellipse is a circle scaled in one direction, it has two axes of symmetry, at right angles to each other. For an ellipse in standard position, the axes of symmetry are the x-axis and the y-axis.The points where an ellipse crosses its axes of symmetry are called its vertices.

22. [Finding the eccentricity of an ellipse in standard position] What's the eccentricity of this ellipse?

23. Justify existence of the second focus and directrix in an ellipse.

    Just because it's symmetric! The symmetry of an ellipse might lead you to suspect that it has another focus and another directrix, and this is indeed the case.By the symmetry of the ellipse, for any point P on the ellipse, its reflection, Q say, in the y-axis also lies on the ellipse. Since the distance of Q from F is e times the distance of Q from d, it follows, again by the symmetry of the ellipse, that the distance of P from F2 is e times the distance of P from d2. Hence F2 is a focus and d2…

24. Sketch approximate ellipses with eccentricity = 0.4, and eccentricity = 0.9.

    Three ellipses and their eccentricities The Earth’s orbit around the Sun is nearly circular, with eccentricity e ≈ 0.0167, whereas the orbit of Halley's comet is elongated, with eccentricity e ≈ 0.967.

25. What's a and b in the equation of an ellipse?

    a - x-intercept b - y-intercept

26. Given this equation - how long is the major axis?

    2a.

27. [Sketching an ellipse in standard position] Consider the equation 9x2 + 16y2 = 144. (a) Show that this equation represents an ellipse in standard position. (b) Find the vertices of the ellipse and sketch it. (c) Find the eccentricity, foci and directrices of the ellipse.

28. How is a circle a special type of ellipse?

    Circle: an ellipse with e=0, and whose foci coincide. It's sometimes useful to think of a circle as a special type of ellipse, whose eccentricity is zero and whose foci coincide. However a circle doesn’t have a focus–directrix property.

29. State equations of a circle VS ellipse.

    Equation of an ellipse: a - x interceptb - y intercept Equation of a circle: a - radiusWhere center is at the origin.

30. In what way is ellipse a scaled circle? Ellipse: Circle:

31. Prove that PF1 + PF2 = 2a for any point on an ellipse.

32. Describe the Gardener's method for drawing ellipses. This method of drawing an ellipse is known as the gardener’s method, since it’s a convenient way to mark out elliptical flower beds. It has been known since at least the sixth century ad, when it was described by Anthemius, the architect of the great church of Hagia Sophia in Constantinople.

    You stick two pins through a piece of paper on a corkboard, say, and attach the ends of a length of string to the pins. Then you use a pencil to hold the string taut, and move the pencil around, with the string always held taut, to trace out a curve.The curve that you obtain is an ellipse whose foci are the positions of the pins.

33. [Planning an elliptical flower bed] A garden design includes an elliptical flower bed 4 metres long and 3 metres wide. The ellipse is to be marked out by the gardener's method, using a length of string and two pegs. (a) What length of string is needed? (b) How far apart should the pegs be placed?

34. Name the interesting feature of the elliptical reflector. An elliptical reflector (also known as an ellipsoidal reflector ) is a reflective surface whose shape is formed by rotating all or part of an ellipse through 360◦ about its major axis. Any cross-section of an elliptical reflector that passes through this major axis is therefore part of an ellipse with the same foci as the original ellipse. We refer to these foci as the foci of the elliptical reflector.

    If a light source is positioned at one focus, then the rays of light are reflected to pass through the other focus. It can be proved, by using the focus–directrix property of an ellipse, that if a light source is positioned at one of the foci of an elliptical reflector (with a surface designed to reflect light), then the rays of light are reflected to pass through the other focus.

35. What's this?

    A sundial. Consists of a flat plate (the dial) and a gnomon, which casts a shadow onto the dial. As the Sun appears to move across the sky, the shadow aligns with different hour-lines, which are marked on the dial to indicate the time of day.

36. What's gnomon?

    A gnomon is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields. A horizontal dial commissioned in 1862, the gnomon is the triangular blade.Gnomon (/ˈnoʊmɒn, ˈnoʊmən/, from Greek γνώμων, gnōmōn, literally: "one that knows or examines")

37. What curves do the sundials' gnomons follow?

    Hyperbolas. The tip of the shadow of the gnomon on a sundial follows a path that is part of a hyperbola.

38. What's a and b in the equation of a hyperbola?

    a - x-intercept b - affects the shape of the hyperbola Since the hyperbola lies between its asymptotes y = ±(b/a)x, the smaller the value of b/a is, the more squashed the hyperbola is vertically. Unlike with ellipses, the values b and −b aren’t y-intercepts of the hyperbola with equation x2/a2 − y2/b2 = 1. The hyperbola doesn’t intersect the y-axis at all.However the value of b affects the shape of the hyperbola in a similar way to the way that the value of b affects the shape of an ellipse in…

39. Judging by the formula of the hyperbola - why no point (x, y) with −a < x < a lies on the hyperbola?

40. What's a rectangular hyperbola?

    Hyperbola whose asymptotes are perpendicular. A rectangular hyperbola in standard position

41. What's the centre of the hyperbola?

    The point where axes of symmetry cross. The point where the axes of symmetry of a hyperbola cross (which is also the point where its asymptotes cross) is called its center.

42. What are the vertices of the hyperbola?

    The 2 points where a hyperbola crosses an axis of symmetry are called its vertices.

43. Sketch approximate hyperbolas with eccentricity = 1.1, and eccentricity = 3.

44. Prove that these lines are the asymptotes of the hyperbola. Remember that a line that a curve approaches more and more closely the further it is from the origin is called an asymptote of the curve.

45. [Identifying types of conics in standard position from their equations] Identify the type of conic.

    (a) hyperbola (b) parabola (c) ellipse

46. What's the general equation for a conic?

    In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section (though it may be degenerate), and all conic sections arise in this way.

47. Name 2 examples where general equation of a conic doesn't produce a conic . Note that not every equation of the form in the box above is the equation of a conic.

    doesn't represent any curve at all: x2 + y2 + 1 = 0 straight line: x + y + 1 = 0

48. How is the hyperbola y = 1/x obtained from the hyperbola in a standard position?

    By rotating x2 - y2 = 2 anticlockwise about the origin through 45°.

49. Provided the general equation does represent a conic - how to determine whether general equation represents ellipse/circle parabola hyperbola?

50. [Determining the type of a conic] Given that the equation represents a non-degenerate conic, determine what type of conic it is.

51. How can you describe a curve in terms of a parameter t without explicitly stating a pair of parametric equations?

    For example, you can say that a particular line or curve is given by the set of all points of this form.

52. What's the name of this curve?

    Hypotrochoid. A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle. The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

53. [Eliminating the parameter from parametric equations] Eliminate the parameter from the parametric equations Eliminating the parameter means using the parametric equations to work out an equation that expresses y in terms of x.

54. [Finding the closest approach of two ships] How would you go about finding the closest distance to which the ships approach each other?

    Express d in terms of t Complete the square

55. [Justify parametrisation] Justify this parametrisation of a straight line. In general, any straight line has infinitely many parametrisations. However, two are particularly useful.

    You can reach any point on the line by placing your pen tip at the point (x0, y0) and moving it t units to the right, and mt units up, for some number t.

56. [Finding a parametrisation of a straight line from a point on it and its gradient] Write down parametric equations for the straight line that passes through the point (−3, 1) and has gradient −2.

57. [Justify parametrisation] Justify this parametrisation of a straight line. In general, any straight line has infinitely many parametrisations. However, two are particularly useful.

    You can reach any point (x, y) on the line by placing your pen tip at the point (x0, y0) and moving it t(x1 − x0) units to the right, and t(y1 − y0) units up, for some number t.

58. [Finding parametrisations for straight lines from two points on them] Write down a parametrisation of the line segment that joins the points (−2, 4) and (3, 1).

    x = -2 + 5t y = 4 - 3t t ∈ [0, 1]

59. [Justify parametrisation] Justify this parametrisation of a circle.

    In the unit circle x = cos(t), y = sin(t) Scale by r Translate by (p, q)

60. [Finding a parametrisation of a semicircle] Write down a parametrisation for the semicircle with centre (1, 1) shown here.

61. [Justify parametrisation] Justify this parametrisation of a parabola, given the formula of the parabola is y2 = 4ax.

    Let's set y = t, because for each value of y, this equation gives exactly one value of x. Let's set y = 2at though, for even prettier output.

62. [Parametrising a parabola in standard position]

63. [Justify parametrisation] Justify this parametrisation of an ellipse.

    We know that x = r cos(t) & y = r sin(t) for a circle (by definitions of sine and cosine). We know that ellipse is a circle scaled by b/a vertically. Ellipse is described by x = a cos(t), y = b sin(t).

64. [Parametrising an ellipse] Write down the standard parametrisation of the ellipse with equation:

65. [Parametrising a hyperbola in standard position] Write down the standard parametrisation of the hyperbola with equation

66. [Parametrising translated curves] Find parametrisation of the line described below.