X2 T03 05 rectangular hyperbola (2010)

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

 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

 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 Q : x  s 2 y  4s

 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 Q : x  s 2 y  4s t 2  s 2 y  4t  4 s

 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 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

 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 2 y  4t  4 s t  s t  s  y  4t  s  4 y 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

 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

X2 T03 05 rectangular hyperbola (2010)

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