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# Conic sections

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### Conic sections

1. 1. REVISITING…… Conic Sections
2. 2. Conic Sections (1) Circle A circle is formed when   2 i.e. when the plane  is perpendicular to the axis of the cones.
3. 3. Conic Sections (2) Ellipse An ellipse is formed when     2 i.e. when the plane  cuts only one of the cones, but is neither perpendicular to the axis nor parallel to the a generator.
4. 4. Conic Sections (3) Parabola A parabola is formed when   i.e. when the plane  is parallel to a generator.
5. 5. Conic Sections (4) Hyperbola A hyperbola is formed when 0   i.e. when the plane  cuts both the cones, but does not pass through the common vertex.
6. 6. CIRCLE A circle is the locus of a variable point on a plane so that its distance (the radius)remains constant from a fixed point (the centre). y P(x,y)  O x
7. 7. DIFFERENT FORMS OF EQUATIONS OF CIRCLE  × The standard equation of circle: ( x  h)2  ( y  k )2  r 2 where (h, k )is the centre of the circle and r is its radius. The parametric equation of a circle: x  r cos  , × y  r sin  The general equation of a circle: x2  y 2  2 gx  2 fy  c  0 where ( g ,  f ) g2  f 2  c is the centre of the circle and is its radius …..
8. 8. 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). y P(x,y) M(-a,0) O focus F(a,0) x
9. 9. Form the definition of parabola, PF = PN ( x  a)  y  x  a 2 2 ( x  a)  y  ( x  a) 2 2 2 x  2ax  a  y  x  2ax  a 2 2 2 2 2 y  4ax 2 standard equation of a parabola
10. 10. vertex axis of symmetry latus rectum (LL’) mid-point of FM = the origin (O) = vertex length of the latus rectum =LL`= 4a
11. 11. Other forms of Parabola y  4ax 2
12. 12. Other forms of Parabola x  4ay 2
13. 13. Other forms of Parabola x  4ay 2
14. 14. 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.
15. 15. 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.
16. 16. 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.
17. 17. DIFFERENT FORMS OF EQUATIONS OF PARABOLA  The standard equation of parabola: 2 ( y  k )  4a( x  h) k where F (a, 0) the focus and (h,is)the vertex of is parabola. × The parametric equation of a parabola: x  at 2 , × y  2at The general equation of a parabola: ax2  by 2  2 gx  2 fy  c  0 with either a=0 or b=0 but both not zero at the same time.
18. 18. 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)
19. 19. Let PF1+PF2 = 2a where a > 0 ( x  c)  y  ( x  c)  y  2a 2 2 2 2 ( x  c)  y  2a  ( x  c)  y 2 2 2 2 ( x  c)  y  4a  4a ( x  c)  y  ( x  c)  y 2 2 2 2 2 4a ( x  c) 2  y 2  4cx  4a 2 a ( x  2 xc  c  y )  c x  2a cx  a 2 2 2 2 2 2 2 4 a 2 x 2  2a 2 xc  a 2c 2  a 2 y 2  c 2 x 2  2a 2cx  a 4 2
20. 20. (a  c ) x  a y  a  a c 2 2 2 2 2 4 2 2 (a  c ) x  a y  a (a  c ) 2 2 2 2 Let b  a  c 2 2 2 2 2 2 2 b x a y a b 2 2 2 2 2 2 2 2 x y  2 1 2 a b standard equation of an ellipse
21. 21. 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 2b 2 length of lactus rectum = a
22. 22. 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
23. 23. 12.4 Equations of an Ellipse
24. 24. 12.4 Equations of an Ellipse Other form of Ellipse 2 2 x y  2 1 2 b a where a2 – b2 = c2 and a > b > 0
25. 25. 12.4 Equations of an Ellipse Furthermore, x2 y2 (1) Given an ellipse 2  2  1, where a  b  0, a b the length of the semi - major axis is a and that of the semi - minor axis is b. x2 y2 (2) Given an ellipse 2  2  1, where a  b  0, b a then its foci lie on the y - axis, the length of the semi - major axis is a and that of the semi - minor axis is b.
26. 26. 12.4 Equations of an Ellipse ( x  h) ( y  k ) (3) The equation   1, represent an 2 2 a b ellipse whose centre is at (h, k ) and whose axes are 2 2 parallel to the coordinate axes. y ( x  h) 2 ( y  k ) 2  1 2 2 a b (h, k) O x
27. 27. DIFFERENT FORMS OF EQUATIONS OF ELLIPSE  The standard equation of ellipse: x2 y 2  2  1, a  b and c 2  a 2  b2 2 a b where F (c, 0) the foci of the ellipse. are × The parametric equation of an ellipse: x  a cos  , y  b sin 
28. 28. 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 two points remains a constant. The two fixed points are called foci. P’(x,y)
29. 29. Let |PF1-PF2| = 2a where a > 0 | ( x  c)  y  ( x  c)  y | 2a 2 2 2 2 ( x  c)  y  2a  ( x  c)  y 2 2 2 2 ( x  c)  y  4a  4a ( x  c)  y  ( x  c)  y 2 2 2 2 2  4a ( x  c) 2  y 2  4cx  4a 2 a ( x  2 xc  c  y )  c x  2a cx  a 2 2 2 2 2 2 2 4 a 2 x 2  2a 2 xc  a 2c 2  a 2 y 2  c 2 x 2  2a 2cx  a 4 2
30. 30. (c  a ) x  a y  a c  a 2 2 2 2 2 2 2 4 (c  a ) x  a y  a (c  a ) 2 2 2 2 Let b  c  a 2 2 2 2 2 2 2 b x a y  a b 2 2 2 2 2 2 2 2 x y  2 1 2 a b standard equation of a hyperbola
31. 31. transverse axis vertex lactus rectum conjugate axis 2b 2 length of lactus rectum = a length of the semi-transverse axis = a length of the semi-conjugate axis = b
32. 32. 12.7 Equations of a Hyperbola A1, A2 vertices A1A2 transverse axis YY’ conjugate axis O centre GH focal chord CD lactus rectum
33. 33. 12.7 Equations of a Hyperbola asymptote b equation of asymptote : y   x a
34. 34. 12.7 Equations of a Hyperbola Other form of Hyperbola : 2 2 y x  2 1 2 a b
35. 35. Rectangular Hyperbola If b = a, then 2 2 x y  2 1 2 a b 2 2 y x  2 1 2 a b x y a 2 2 2 y x a 2 2 2 The hyperbola is said to be rectangular hyperbola.
36. 36. equation of asymptote : x y 0
37. 37. 12.7 Equations of a Hyperbola Properties of a hyperbola : ( x - h) 2 ( y - k ) 2 (1) The equation   1 represents a 2 2 a b hyperbola with centre at (h, k ), transverse axis parallel to the x - axis. ( x - h) 2 ( y - k ) 2 (2) The equation   1 represents a 2 2 a b hyperbola with centre at (h, k ), transverse axis parallel to the y - axis.
38. 38. 12.7 Equations of a Hyperbola Parametric form of a hyperbola :  x  a sec    y  b tan  where  is a parameter.  the point (a sec  , b tan  ) lies on the 2 2 x y hyperbola 2  2  1. a b
39. 39. 12.8 Asymptotes of a Hyperbola 2 2 x y The hyperbola 2  2  1, where a, b a b are positive constants, has two asymptotes x y   0. a b
40. 40. 12.8 Asymptotes of a Hyperbola Properties of asymptotes to a hyperbola : x2 y2 (1) The hyperbola - 2  2  1 has two asymptotes a b x y    0. a b ( x  h) 2 ( y  k ) 2 (2) The hyperbola   1 has two 2 2 a b xh yk asymptotes   0. a b
41. 41. 12.8 Asymptotes of a Hyperbola Properties of asymptotes to a hyperbola : ( x  h) ( y  k ) (3) The hyperbola    1 has two 2 2 a b xh yk asymptotes    0. a b 2 2
42. 42. Simple Parametric Equations and Locus Problems x = f(t) y = g(t) parametric equations parameter Combine the two parametric equations into one equation which is independent of t. Then sketch the locus of the equation.
43. 43. Equation of Tangents to Conics general equation of conics : Ax  Bxy  Cy  Dx  Ey  F  0 2 2 Steps : dy (1) Differentiate the implicit equation to find . dx dy (2) Put the given contact point (x1, y1) into dx to find out the slope of tangent at that point. (3) Find the equation of the tangent at that point.
44. 44. THE GENERAL EQUATION OF SECOND DEGREE Ax 2  By 2  Gx  Fy  C  0  Case I: IfA  B  0, the equation represents a circle with centre G F G ( , at 2 A 2 A ) and radius 4 A  4FA  C A Case II: If A  Band both have the same sign, the equation represents the standard equation of an ellipse in XY-coordinate G F X  x and Y  y  system, where 2A 2B Case III: If A  B and both have opposite signs, the equation represents the standard equation of hyperbola in XY-coordinate G F system, where X  x  2 A and Y  y  2( B) Case IV: If A  0 or B  0 ,the equation represents the standard equation of parabola in XY- coordinate system, where 2 2    2 2 G C G2 X  x and Y  y   2A F 4 AF
45. 45. THE DISCRIMINANT TEST With the understanding that occasional degenerate cases may arise, the quadratic curve Ax2  Bxy  Cy 2  Dx  Ey  F  0 is 2  a parabola, if B  4 AC  0 2  an ellipse, if B  4 AC  0 2  a hyperbola, if B  4 AC  0 
46. 46. CLASSIFYING CONIC SECTION BY ECCENTRICITY      In both ellipse and hyperbola, the eccentricity is the ratio of the distance between the foci to the distance between the vertices. Suppose the distance PF of a point P from a fixed point F (the focus)is a constant multiple of its distance from a fixed line (the directrix).i.e. PF  e.PD where e is the constant of , proportionality. Then the path traced by P is (a). a parabola if e  1 (b). an ellipse of eccentricity e if e  1 (c). a hyperbola of eccentricity e if e  1
47. 47. Conics Parabola Ellipse Hyperbola PF = PN PF1 + PF2 = 2a | PF1 - PF2 | = 2a Graph Definitio n
48. 48. Conics Parabola Ellipse Hyperbola x2 y2  2 1 2 a b x2 y2  2 1 2 a b Graph Standard Equation y  4ax 2
49. 49. Conics Parabola Ellipse Hyperbola x = -a a x  , e  PF1 e PN a PF1 x  ,e PN e Graph Directrix
50. 50. Conics Parabola Ellipse Hyperbola Graph Vertices (0,0) A(-a,0), B(a,0), C(0,b), D(0,-b) A1(a,0), A2(-a,0)
51. 51. Conics Parabola Ellipse Hyperbola major axis = AB minor axis =CD transverse axis =A1A2 conjugate axis =B1B2 where B1(0,b), B2(0,-b) Graph Axes axis of parabola = the x-axis
52. 52. Conics Parabola Ellipse Hyperbola 4a 2b 2 a 2b 2 a Graph Length of lantus rectum LL`
53. 53. Conics Parabola Ellipse Hyperbola ---- ---- b y x a Graph Asymptotes
54. 54. Conics Parabola Ellipse Hyperbola Graph Parametric representation of P 2 (at ,2at ) (a cos  , b sin  ) (a sec , b tan )