polar coordinates practice...A curve C1 has polar equation r = 2sin θ, 0 2≤

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COORDINATES

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CONVERTING BETWEEN

CARTESIANS AND POLARS

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Question 1

A curve C has Cartesian equation

( ) ( )2

2 2 2 2 2x y a x y+ = − , 0a ≠ .

Determine a polar equation for C .

2 2 cos 2r a θ=

Question 2

( ) ( )3 42 2

x y x y+ = + .

Show that a polar equation for C is given by

1 sin 2r θ= + , 0r ≥ .

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Question 3

2 2 22x y x y+ = .

Show that a polar equation for C can be written as

sin 2 cosr θ θ= .

Question 4

A circle has Cartesian equation

( ) ( )2 2

3 4 25x y− + − = .

Show that a polar equation for the circle is given by

cos sinr A Bθ θ= + ,

where A and B are constants.

6cos 4sinr θ θ= −

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Question 5

A circle has polar equation

( )4 cos sinr θ θ= + 0 2θ π≤ < .

Determine the Cartesian coordinates of the centre of the circle and the length of its

radius.

( )2,2 , radius 8=

Question 6

Write the polar equation

cos sinr θ θ= + , 0 2θ π≤ <

in Cartesian form, and hence show that it represents a circle, further determining the

coordinates of its centre and the size of its radius.

( ) ( )2 2

1 1 12 2 2

x y− + − =

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Question 7

A curve C has polar equation

1 cos 2r θ= + .

Determine a Cartesian equation for C .

2 2 44x y x+ =

Question 8

sec tanr θ θ= + .

Determine a Cartesian equation for C .

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Question 9

1 cosr

+, 0 2θ π≤ < .

a) Find a Cartesian equation for C .

b) Sketch the graph of C .

( )2 4 1y x= −

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Question 10

The curve C has Cartesian equation

( )( )22 2 21x y x x+ − = .

Find a polar equation of C in the form ( )r f θ= .

1 secr θ= +

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Question 11

A curve 1C has polar equation

2sinr θ= , 0 2θ π≤ < .

a) Find a Cartesian equation for 1C , and describe it geometrically.

A different curve 2C has Cartesian equation

−, 1x ≠ ± .

b) Find a polar equation for 2C , in the form ( )r f θ= .

( )22 1 1x y+ − = , tanr θ=

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Question 12

Show that the polar equation of the top half of the parabola with Cartesian equation

2 1y x= + , 12

x ≥ − ,

is given by the polar equation

1 cosr

−, 0r ≥ .

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Question 13

The points A and B have respective coordinates ( )1,0− and ( )1,0 . The locus of the

point ( ),P x y traces a curve in such a way so that 1AP BP = .

a) By forming a Cartesian equation of the locus of P , show that the polar

equation of the curve is

2 2cos2r θ= , 0 2θ π≤ < .

b) Sketch the curve.

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Question 14

The curve C has polar equation

tanr θ= , 02

πθ≤ < .

Find a Cartesian equation of C in the form ( )y f x= .

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TANGENTS TO POLAR CURVES

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Question 1

A Cardioid has polar equation

1 2cosr θ= + , 02

πθ≤ ≤ .

The point P lies on the Cardioid so that the tangent to the Cardioid at P is parallel to

the initial line.

Determine the exact length of OP , where O is the pole.

( )1 3 334

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Question 2

The figure above shows the polar curve with equation

sin 2r θ= , 02

πθ≤ ≤ .

The point P lies on the curve so that the tangent at P is parallel to the initial line.

Find the Cartesian coordinates of P .

( )2 46, 39 9

initial line

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POLAR CURVE AREAS

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Question 1

The figure above shows a spiral curve with polar equation

r aθ= , 0 2θ π≤ ≤ ,

where a is a positive constant.

Find the area of the finite region bounded by the spiral and the initial line.

2 34area3

r aθ=

initial lineO

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Question 2

The figure above shows the polar curve C with equation

sin 2r θ= , 02

πθ≤ ≤ .

Find the exact value of the area enclosed by the curve.

area8π=

initial line

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Question 3

The figure above shows a circle with polar equation

( )4 cos sinr θ θ= + 0 2θ π≤ < .

Find the exact area of the shaded region bounded by the circle, the initial line and the

half line 2

θ π= .

area 4 8π= +

2θ π=

initial line

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Question 4

The polar curve C has equation

2 cosr θ= + , 0 2θ π≤ < .

a) Sketch the graph of C .

b) Show that the area enclosed by the curve is 9

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Question 5

The figure above shows the polar curve C with equation

2sin 2 cosr θ θ= , 2 2

π πθ− ≤ ≤ .

Show that the area enclosed by one of the two identical loops of the curve is 16

O initial line

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Question 6

The diagram above shows the curve with polar equation

2sinr a θ= + , 0 2θ π≤ < ,

where a is a positive constant.

Determine the value of a given that the area bounded by the curve is 38π .

2sinr a θ= +

initial lineO

polar coordinates practice...A curve C1 has polar equation r = 2sin θ, 0 2≤

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