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Unit 12

Unit 12. Conic Sections. A circle is formed when i.e. when the plane  is perpendicular to the axis of the cones. Conic Sections. (1) Circle. An ellipse is formed when

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Unit 12

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  1. Unit 12 Conic Sections

  2. A circle is formed when i.e. when the plane  is perpendicular to the axis of the cones. Conic Sections (1) Circle

  3. An ellipse is formed when i.e. when the plane  cuts only one of the cones, but is neither perpendicular to the axis nor parallel to the a generator. Conic Sections (2) Ellipse

  4. A parabola is formed when i.e. when the plane  is parallel to a generator. Conic Sections (3) Parabola

  5. A hyperbola is formed when i.e. when the plane  cuts both the cones, but does not pass through the common vertex. Conic Sections (4) Hyperbola

  6. y P(x,y) M(-a,0) O focus F(a,0) x Parabola A parabola is the locus of a variable point on a plane so that its distance from a fixed point (the focus) is equal to its distance from a fixed line (the directrix x = - a).

  7. Form the definition of parabola, PF = PN standard equation of a parabola

  8. axis of symmetry vertex latus rectum (LL’) mid-point of FM = the origin (O) = vertex length of the latus rectum = LL’= 4a

  9. Other forms of Parabola

  10. Other forms of Parabola

  11. Other forms of Parabola

  12. 12.1 Equations of a Parabola A parabola is the locus of a variable point P which moves in a plane so that its distance from a fixed point F in the plane equals its distance from a fixed line l in the plane. The fixed point F is called the focus and the fixed line l is called the directrix.

  13. 12.1 Equations of a Parabola The equation of a parabola with focus F(a,0) and directrix x + a =0, where a >0, is y2 = 4ax.

  14. 12.1 Equations of a Parabola X’X is the axis. O is the vertex. F is the focus. MN is the focal chord. HK is the latus rectum.

  15. Equation of the tangent to the parabola y2 = 4ax at the point (x1, y1) is y1y = 2a(x + x1), and that the point (at2, 2at) is 12.2 Chords and Tangents of a Parabola

  16. 12.2 Chords and Tangents of a Parabola

  17. 12.2 Chords and Tangents of a Parabola

  18. 12.2 Chords and Tangents of a Parabola A straight line, not parallel to the axis of a parabola, is a tangent to the parabola if and only if it cuts the parabola at exactly one point.

  19. 12.2 Chords and Tangents of a Parabola Remarks : In general, a tangent to a curve may not cut the curve at exactly one point. In parabola, the situation is quite different. A tangent to the parabola does not meet the curve in addition to the point of contact. This fact enables us to find the equation of the tangent algebraically. P Q

  20. 12.2 Chords and Tangents of a Parabola Let P be a point not on the parabola such that two tangents, touching the curve at two points respectively, can be drawn from P to the parabola. The chord joining the points of contact is called the chord of contact of tangents drawn to the parabola from the point P.

  21. 12.2 Chords and Tangents of a Parabola The equation of the chord of contact of tangents drawn to the parabola y2 = 4ax from the point P(x1, y1) is y1y = 2a(x + x1).

  22. 12.3 Normal to a Parabola

  23. P.441 Ex.12A

  24. 12.4 Equations of an Ellipse An ellipse is a curve which is the locus of a variable point which moves in a plane so that the sum of its distance from two fixed points remains a constant. The two fixed points are called foci. P’(x,y) P’’(x,y)

  25. Let PF1+PF2 = 2a where a > 0

  26. standard equation of an ellipse

  27. length of lactus rectum = 12.4 Equations of an Ellipse major axis = 2a vertex lactus rectum minor axis = 2b length of semi-major axis = a length of the semi-minor axis = b

  28. 12.4 Equations of an Ellipse AB major axis CD minor axis A, B, C and D vertices O centre PQ focal chord F focus RS, R’S’ latus rectum

  29. 12.4 Equations of an Ellipse

  30. 12.4 Equations of an Ellipse Other form of Ellipse where a2 – b2 = c2 and a > b > 0

  31. 12.4 Equations of an Ellipse

  32. 12.4 Equations of an Ellipse y (h, k) x O

  33. 12.5 Chords and Tangents of an Ellipse

  34. 12.5 Chords and Tangents of an Ellipse

  35. 12.5 Chords and Tangents of an Ellipse

  36. 12.5 Chords and Tangents of an Ellipse A straight line is a tangent to an ellipse if and only if it cuts (touches) the ellipse at exactly one point.

  37. 12.6 Normal to an Ellipse

  38. P.453 Ex.12B

  39. 12.7 Equations of a Hyperbola A hyperbola is a curve which is the locus of a variable point which moves in a plane so that the difference of its distance from it to two points remains a constant. The two fixed points are called foci. P’(x,y)

  40. Let |PF1-PF2|= 2a where a > 0

  41. standard equation of a hyperbola

  42. length of lactus rectum = transverse axis vertex lactus rectum conjugate axis length of the semi-transverse axis = a length of the semi-conjugate axis = b

  43. 12.7 Equations of a Hyperbola A1, A2 vertices A1A2 transverse axis YY’ conjugate axis O centre GH focal chord CD lactus rectum

  44. 12.7 Equations of a Hyperbola asymptote equation of asymptote :

  45. 12.7 Equations of a Hyperbola Other form of Hyperbola :

  46. Rectangular Hyperbola If b = a, then The hyperbola is said to be rectangular hyperbola.

  47. equation of asymptote :

  48. 12.7 Equations of a Hyperbola Properties of a hyperbola :

  49. 12.7 Equations of a Hyperbola Parametric form of a hyperbola :

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