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X2 T03 05 rectangular hyperbola (2011)

N
Nigel Simmons

The document discusses the properties of a rectangular hyperbola. A rectangular hyperbola is a hyperbola whose asymptotes are perpendicular to each other. The equation of a rectangular hyperbola in standard form is x2/a2 - y2/a2 = 1, where the foci are located at (�a, 0) and the distance between the directrices is 2a. Rotating the hyperbola 45 degrees anticlockwise transforms it into the form x-y=�a2, with the foci now located at (a, a).

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Rectangular Hyperbola
Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other
Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a
Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba
Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba  hyperbola has the equation; x2 y2 2  2 1 a a x2  y2  a2
Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba  hyperbola has the equation; a 2 e 2  1  a 2 x2 y2 2  2 1 a a x2  y2  a2
Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba  hyperbola has the equation; a 2 e 2  1  a 2 x2 y2 e2  1  1 2  2 1 a a e2  2 x2  y2  a2 e 2
Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba  hyperbola has the equation; a 2 e 2  1  a 2 x2 y2 e2  1  1 2  2 1 a a e2  2 x2  y2  a2 e 2  eccentricity is 2
y Y P  x, y  x X
y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X
y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45
y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45    x  iy  1 1   i   2 2
y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45    x  iy  1 1   i   2 2 1   x  iy 1  i  2 1   x  ix  iy  y  2 x y x y   i 2 2
y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45   1 i 1  x y x y   x  iy   X  Y  2 2 2 2 1   x  iy 1  i  2 1   x  ix  iy  y  2 x y x y   i 2 2
y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45   1 i 1  x y x y   x  iy   X  Y  2 2 2 2 1 x2  y2   x  iy 1  i  XY  2 2 1   x  ix  iy  y  2 x y x y   i 2 2
y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45   1 i 1  x y x y   x  iy   X  Y  2 2 2 2 1 x2  y2   x  iy 1  i  XY  2 2 1 a2   x  ix  iy  y  XY  2 2 x y x y   i 2 2
focus;  ae,0    2a,0 
focus;  ae,0    2a,0   1  1 i 2a   2 2   a  ai
focus;  ae,0    2a,0   1  1 i 2a   2 2   a  ai  focus a, a 
a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a   2 2   a  ai  focus a, a 
a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai  focus a, a 
a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a 
a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2
a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2 a  distance from origin to directrix is 2
a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2 a  distance from origin to directrix is 2 00k a  2 2
a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2 a  distance from origin to directrix is 2 00k a  2 2 k a k  a
a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2 a  distance from origin to directrix is 2 00k a  2 2 k a k  a  directrices are x  y   a
The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2
The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2 Parametric Coordinates of xy  c 2
The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2 Parametric Coordinates of xy  c 2 c x  ct y t
The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2 Parametric Coordinates of xy  c 2 c x  ct y t Tangent: x  t 2 y  2ct
The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2 Parametric Coordinates of xy  c 2 c x  ct y t Tangent: x  t 2 y  2ct Normal: t 3 x  ty  ct 4  1
e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry.
e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x x
e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2
e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t
e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t 4 y x dy 4  2 dx x
e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t 4 dy 4 y when x  2t ,   x dx 2t 2 dy 4  2 1  2 dx x t
e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t 4 dy 4 2 1 y when x  2t ,   y    2  x  2t  x dx 2t 2 t t dy 4  2 1  2 dx x t
e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t 4 dy 4 2 1 y when x  2t ,   y    2  x  2t  x dx 2t 2 t t dy 4 1 t 2 y  2t   x  2t  2  2 dx x t x  t 2 y  4t
 s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st
 s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t Q : x  s 2 y  4s
 s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t Q : x  s 2 y  4s t 2  s 2 y  4t  4 s
 s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t Q : x  s 2 y  4s t 2  s 2 y  4t  4 s t  s t  s  y  4t  s  4 y st
 s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t 4t 2 x  4t Q : x  s 2 y  4s st t 2  s 2 y  4t  4 s t  s t  s  y  4t  s  4 y st
 s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t 4t 2 x  4t Q : x  s 2 y  4s st t 2  s y  4t  4 s 2 x 4 st  4t 2  4t 2 st t  s t  s  y  4t  s  4 st 4  y st st
 2 s, 2  c) s  0, s  t , show that the tangents at P and Q  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t 4t 2 x  4t Q : x  s 2 y  4s st t 2  s y  4t  4 s 2 x 4 st  4t 2  4t 2 st t  s t  s  y  4t  s  4 st 4  y st st  4 st , 4   M is    st st
1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin.
1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st
1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1
1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 x st
1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 y 4 x st st  x
1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 y 4 x st st  x  y  x
1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 y 4 x st st  x 4  y  x  0, thus M  0,0  st
1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 y 4 x st st  x 4  y  x  0, thus M  0,0  st  locus of M is y   x, excluding 0,0 
(ii) Show that PS  PS   2a P  x, y  S S x
(ii) Show that PS  PS   2a y P  x, y  S S x By definition of an ellipse;
(ii) Show that PS  PS   2a y P  x, y  M S S x a x x a e e By definition of an ellipse; PS  PS   ePM
(ii) Show that PS  PS   2a y P  x, y  M M S S x a x x a e e By definition of an ellipse; PS  PS   ePM  ePM 
(ii) Show that PS  PS   2a y P  x, y  M M S S x a x x a e e By definition of an ellipse; PS  PS   ePM  ePM   e PM  PM   2a   e   e   2a
(ii) Show that PS  PS   2a y P  x, y  M M S S x a x x a e e By definition of an ellipse; PS  PS   ePM  ePM   e PM  PM  Exercise 6D; 3, 4, 7, 10, 11a,  2a   e  12, 14, 19, 21, 26, 29,  e  31, 43, 47  2a

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X2 T03 05 rectangular hyperbola (2011)

  • 2. Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other
  • 3. Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a
  • 4. Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba
  • 5. Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba  hyperbola has the equation; x2 y2 2  2 1 a a x2  y2  a2
  • 6. Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba  hyperbola has the equation; a 2 e 2  1  a 2 x2 y2 2  2 1 a a x2  y2  a2
  • 7. Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba  hyperbola has the equation; a 2 e 2  1  a 2 x2 y2 e2  1  1 2  2 1 a a e2  2 x2  y2  a2 e 2
  • 8. Rectangular Hyperbola A hyperbola whose asymptotes are perpendicular to each other b b   1 a a b2  a 2 ba  hyperbola has the equation; a 2 e 2  1  a 2 x2 y2 e2  1  1 2  2 1 a a e2  2 x2  y2  a2 e 2  eccentricity is 2
  • 9. y Y P  x, y  x X
  • 10. y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X
  • 11. y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45
  • 12. y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45    x  iy  1 1   i   2 2
  • 13. y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45    x  iy  1 1   i   2 2 1   x  iy 1  i  2 1   x  ix  iy  y  2 x y x y   i 2 2
  • 14. y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45   1 i 1  x y x y   x  iy   X  Y  2 2 2 2 1   x  iy 1  i  2 1   x  ix  iy  y  2 x y x y   i 2 2
  • 15. y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45   1 i 1  x y x y   x  iy   X  Y  2 2 2 2 1 x2  y2   x  iy 1  i  XY  2 2 1   x  ix  iy  y  2 x y x y   i 2 2
  • 16. y Y In order to make the P  x, y  asymptotes the coordinate axes we need to rotate the x curve 45 degrees anticlockwise. X i.e. P x, y   x  iy is multiplied by cis 45  x  iy cos 45  i sin 45   1 i 1  x y x y   x  iy   X  Y  2 2 2 2 1 x2  y2   x  iy 1  i  XY  2 2 1 a2   x  ix  iy  y  XY  2 2 x y x y   i 2 2
  • 17. focus;  ae,0    2a,0 
  • 18. focus;  ae,0    2a,0   1  1 i 2a   2 2   a  ai
  • 19. focus;  ae,0    2a,0   1  1 i 2a   2 2   a  ai  focus a, a 
  • 20. a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a   2 2   a  ai  focus a, a 
  • 21. a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai  focus a, a 
  • 22. a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a 
  • 23. a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2
  • 24. a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2 a  distance from origin to directrix is 2
  • 25. a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2 a  distance from origin to directrix is 2 00k a  2 2
  • 26. a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2 a  distance from origin to directrix is 2 00k a  2 2 k a k  a
  • 27. a focus;  ae,0  directrix; x e   2a,0  a x 2  1  1 i 2a  directrices are || to y axis  2 2   when rotated || to y   x  a  ai thus in form x  y  k  0  focus a, a  2a Now distance between directrices is 2 a  distance from origin to directrix is 2 00k a  2 2 k a k  a  directrices are x  y   a
  • 28. The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2
  • 29. The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2 Parametric Coordinates of xy  c 2
  • 30. The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2 Parametric Coordinates of xy  c 2 c x  ct y t
  • 31. The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2 Parametric Coordinates of xy  c 2 c x  ct y t Tangent: x  t 2 y  2ct
  • 32. The rectangular hyperbola with x and y axes as aymptotes, has the equation; 1 2 xy  a 2 where; foci :  a, a  directrices : x  y   a eccentricity  2 Parametric Coordinates of xy  c 2 c x  ct y t Tangent: x  t 2 y  2ct Normal: t 3 x  ty  ct 4  1
  • 33. e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry.
  • 34. e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x x
  • 35. e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2
  • 36. e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t
  • 37. e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t 4 y x dy 4  2 dx x
  • 38. e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t 4 dy 4 y when x  2t ,   x dx 2t 2 dy 4  2 1  2 dx x t
  • 39. e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t 4 dy 4 2 1 y when x  2t ,   y    2  x  2t  x dx 2t 2 t t dy 4  2 1  2 dx x t
  • 40. e.g. (i) (1991) The hyperbola H is xy= 4 a) Sketch H showing where H intersects the axis of symmetry. y y=x xy  4 2,2 x2  4 x  2,2 x  2  2t , 2  is x  t 2 y  4t b) Show that the tangent at P   t 4 dy 4 2 1 y when x  2t ,   y    2  x  2t  x dx 2t 2 t t dy 4 1 t 2 y  2t   x  2t  2  2 dx x t x  t 2 y  4t
  • 41.  s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st
  • 42.  s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t Q : x  s 2 y  4s
  • 43.  s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t Q : x  s 2 y  4s t 2  s 2 y  4t  4 s
  • 44.  s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t Q : x  s 2 y  4s t 2  s 2 y  4t  4 s t  s t  s  y  4t  s  4 y st
  • 45.  s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t 4t 2 x  4t Q : x  s 2 y  4s st t 2  s 2 y  4t  4 s t  s t  s  y  4t  s  4 y st
  • 46.  s, 2  c) s  0, s  t , show that the tangents at P and Q 2  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t 4t 2 x  4t Q : x  s 2 y  4s st t 2  s y  4t  4 s 2 x 4 st  4t 2  4t 2 st t  s t  s  y  4t  s  4 st 4  y st st
  • 47.  2 s, 2  c) s  0, s  t , show that the tangents at P and Q  2 2  s intersect at M  4 st , 4    st st P : x  t 2 y  4t 4t 2 x  4t Q : x  s 2 y  4s st t 2  s y  4t  4 s 2 x 4 st  4t 2  4t 2 st t  s t  s  y  4t  s  4 st 4  y st st  4 st , 4   M is    st st
  • 48. 1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin.
  • 49. 1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st
  • 50. 1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1
  • 51. 1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 x st
  • 52. 1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 y 4 x st st  x
  • 53. 1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 y 4 x st st  x  y  x
  • 54. 1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 y 4 x st st  x 4  y  x  0, thus M  0,0  st
  • 55. 1 d) Suppose that s  , show that the locus of M is a straight t line through the origin, but not including the origin. 4 st 4 x y st st 1 s t st  1 4 y 4 x st st  x 4  y  x  0, thus M  0,0  st  locus of M is y   x, excluding 0,0 
  • 56. (ii) Show that PS  PS   2a P  x, y  S S x
  • 57. (ii) Show that PS  PS   2a y P  x, y  S S x By definition of an ellipse;
  • 58. (ii) Show that PS  PS   2a y P  x, y  M S S x a x x a e e By definition of an ellipse; PS  PS   ePM
  • 59. (ii) Show that PS  PS   2a y P  x, y  M M S S x a x x a e e By definition of an ellipse; PS  PS   ePM  ePM 
  • 60. (ii) Show that PS  PS   2a y P  x, y  M M S S x a x x a e e By definition of an ellipse; PS  PS   ePM  ePM   e PM  PM   2a   e   e   2a
  • 61. (ii) Show that PS  PS   2a y P  x, y  M M S S x a x x a e e By definition of an ellipse; PS  PS   ePM  ePM   e PM  PM  Exercise 6D; 3, 4, 7, 10, 11a,  2a   e  12, 14, 19, 21, 26, 29,  e  31, 43, 47  2a