X2 t03 05 rectangular hyperbola (2012)

Rectangular Hyperbola

Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other

Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other

1

ab

ab

Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other

1

ab

ab

abab

22

Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other

1

ab

ab

abab

22

equation;thehas hyperbola

222

2

2

2

2

1

ayxay

ax

Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other

1

ab

ab

abab

22

222 1 aea equation;thehas hyperbola

222

2

2

2

2

1

ayxay

ax

Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other

1

ab

ab

abab

22

222 1 aea equation;thehas hyperbola

222

2

2

2

2

1

ayxay

ax

2211

2

2

ee

e

Rectangular HyperbolaA hyperbola whose asymptotes are perpendicular to each other

1

ab

ab

abab

22

222 1 aea equation;thehas hyperbola

222

2

2

2

2

1

ayxay

ax

2211

2

2

ee

e

2 isty eccentrici

y

x

Y

X

yxP ,

y

x

Y

X

yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.

y

x

Y

X

yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.

45cisby multiplied is , i.e. iyxyxP

y

x

Y

X

yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.

45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx

21

21 iiyx

y

x

Y

X

yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.

45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx

21

21 iiyx

iyxyx

yiyixx

iiyx

22

21

12

1

y

x

Y

X

yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.

45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx

21

21 iiyx

iyxyx

yiyixx

iiyx

22

21

12

1

2

2yxYyxX

y

x

Y

X

yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.

45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx

21

21 iiyx

iyxyx

yiyixx

iiyx

22

21

12

1

2

2yxYyxX

2

22 yxXY

y

x

Y

X

yxP ,In order to make the asymptotes the coordinate axes we need to rotate the curve 45 degrees anticlockwise.

45cisby multiplied is , i.e. iyxyxP 45sin45cos iiyx

21

21 iiyx

iyxyx

yiyixx

iiyx

22

21

12

1

2

2yxYyxX

2

22 yxXY

2

2aXY

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0,2a

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21

212

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21

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2200 ak

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2200 ak

akak

ayx aresdirectrice

The rectangular hyperbola with x and y axes as aymptotes, has the equation;

2

21 axy

where; aa , :foci

ayx :sdirectrice

2ty eccentrici

The rectangular hyperbola with x and y axes as aymptotes, has the equation;

2

21 axy

where; aa , :foci

ayx :sdirectrice

2ty eccentrici

2 cxyofsCoordinateParametric

The rectangular hyperbola with x and y axes as aymptotes, has the equation;

2

21 axy

where; aa , :foci

ayx :sdirectrice

2ty eccentrici

2 cxyofsCoordinateParametric

ctx tcy

The rectangular hyperbola with x and y axes as aymptotes, has the equation;

2

21 axy

where; aa , :foci

ayx :sdirectrice

2ty eccentrici

2 cxyofsCoordinateParametric

ctx tcy

Tangent: ctytx 22

The rectangular hyperbola with x and y axes as aymptotes, has the equation;

2

21 axy

where; aa , :foci

ayx :sdirectrice

2ty eccentrici

2 cxyofsCoordinateParametric

ctx tcy

Tangent: ctytx 22 Normal: 143 tctyxt

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

x

y = x

e.g. (i) (1991)The hyperbola H is xy= 4

a) Sketch H showing where H intersects the axis of symmetry.y

x

y = x

244

2

xxxy

2,2

2,2

e.g. (i) (1991)The hyperbola H is xy= 4

a) Sketch H showing where H intersects the axis of symmetry.y

x

y = x

244

2

xxxy

2,2

2,2

tytxt

tP 4 is 2,2at tangent that theShow b) 2

e.g. (i) (1991)The hyperbola H is xy= 4

a) Sketch H showing where H intersects the axis of symmetry.y

x

y = x

244

2

xxxy

2,2

2,2

tytxt

tP 4 is 2,2at tangent that theShow b) 2

24

4

xdxdy

xy

e.g. (i) (1991)The hyperbola H is xy= 4

a) Sketch H showing where H intersects the axis of symmetry.y

x

y = x

244

2

xxxy

2,2

2,2

tytxt

tP 4 is 2,2at tangent that theShow b) 2

24

4

xdxdy

xy

2

2

1

24,2when

t

tdxdytx

e.g. (i) (1991)The hyperbola H is xy= 4

a) Sketch H showing where H intersects the axis of symmetry.y

x

y = x

244

2

xxxy

2,2

2,2

tytxt

tP 4 is 2,2at tangent that theShow b) 2

24

4

xdxdy

xy

2

2

1

24,2when

t

tdxdytx

txtt

y 2122

e.g. (i) (1991)The hyperbola H is xy= 4

a) Sketch H showing where H intersects the axis of symmetry.y

x

y = x

244

2

xxxy

2,2

2,2

tytxt

tP 4 is 2,2at tangent that theShow b) 2

24

4

xdxdy

xy

2

2

1

24,2when

t

tdxdytx

txtt

y 2122

tytxtxtyt

422

2

2

tstsstM

ssQPtss

4,4at intersect

2,2 and at tangents that theshow ,,0 c) 22

tstsstM

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4,4at intersect

2,2 and at tangents that theshow ,,0 c) 22

tytxP 4: 2 sysxQ 4: 2

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4,4at intersect

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tytxP 4: 2 sysxQ 4: 2

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tstsstM

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4,4at intersect

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tytxP 4: 2 sysxQ 4: 2

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tsy

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4

4

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tytxP 4: 2 sysxQ 4: 2

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4

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tstsstM

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tytxP 4: 2 sysxQ 4: 2

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tsy

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4

4

tts

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tsst

tsttstx

4

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tstsstM

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4,4at intersect

2,2 and at tangents that theshow ,,0 c) 22

tytxP 4: 2 sysxQ 4: 2

styst 4422

tsy

stystst

4

4

tts

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tsst

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4

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tstsstM 4,4 is

origin. theincludingnot but origin, he through tline

straight a is of locus that theshow,1 that Suppose d) Mt

s

origin. theincludingnot but origin, he through tline

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s

tsy

tsstx

4 4

origin. theincludingnot but origin, he through tline

straight a is of locus that theshow,1 that Suppose d) Mt

s

tsy

tsstx

4 4

ts 1

1st

origin. theincludingnot but origin, he through tline

straight a is of locus that theshow,1 that Suppose d) Mt

s

tsy

tsstx

4 4

ts 1

4xs t

1st

origin. theincludingnot but origin, he through tline

straight a is of locus that theshow,1 that Suppose d) Mt

s

tsy

tsstx

4 4

ts 1

4xs t

4ys t

x

1st

origin. theincludingnot but origin, he through tline

straight a is of locus that theshow,1 that Suppose d) Mt

s

tsy

tsstx

4 4

ts 1

4xs t

4ys t

x

xy

1st

origin. theincludingnot but origin, he through tline

straight a is of locus that theshow,1 that Suppose d) Mt

s

tsy

tsstx

4 4

ts 1

4xs t

4ys t

x

xy 0,0 thus,04

M

ts

1st

origin. theincludingnot but origin, he through tline

straight a is of locus that theshow,1 that Suppose d) Mt

s

tsy

tsstx

4 4

ts 1

4xs t

4ys t

x

xy 0,0 thus,04

M

ts 0,0excluding,isoflocus xyM

1st

(ii) aSPPS 2 that Show

x

yxP ,

SS

(ii) aSPPS 2 that Show

y

x

yxP ,

SS

By definition of an ellipse;

(ii) aSPPS 2 that Show

y

x

yxP ,

SS

By definition of an ellipse;

ePMSPPS

eax

M

eax

(ii) aSPPS 2 that Show

y

x

yxP ,

SS

By definition of an ellipse;

ePMSPPS MeP

eax

M

eax

M

(ii) aSPPS 2 that Show

y

x

yxP ,

SS

By definition of an ellipse;

ePMSPPS MeP

eax

M

eax

M

MPPMe

aeae

2

2

(ii) aSPPS 2 that Show y

x

yxP ,

SS

By definition of an ellipse;

ePMSPPS MeP

eax

M

eax

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2

2

Exercise 6D; 3, 4, 7, 10, 11a,12, 14, 19, 21, 26, 29,

31, 43, 47

Komarami Unit 3