![1 Find the cartesian equation corresponding to the polar equation r = (
√
2) sec(θ − 1
4
π). [3]
Sketch the the graph of r = (
√
2) sec(θ − 1
4
π), for −1
4
π < θ < 3
4
π, indicating clearly the polar coordinates
of the intersection with the initial line. [2]
Assignment 1
Name:____________________ Group: ___________
Due: Thur, Sep 3rdPolar Coordinates
Notice: Question 4 and 8 wil be marked.](https://image.slidesharecdn.com/assignment1-201021064334/85/Assignment-1-polar-equation-revision-exercise-1-320.jpg)
![The curve C has cartesian equation
(x2
+ y2
)
2
= a2
(x2
− y2
),
where a is a positive constant. Show that C has polar equation
r2
= a2
cos 2θ. [2]
Sketch C for −π < θ ≤ π. [2]
Find the area of the sector between θ = −1
4
π and θ = 1
4
π. [3]
Find the polar coordinates of all points of C where the tangent is parallel to the initial line. [7]
2](https://image.slidesharecdn.com/assignment1-201021064334/85/Assignment-1-polar-equation-revision-exercise-2-320.jpg)
![The curve C has polar equation
r = θ sin θ,
where 0 ≤ θ ≤ π. Draw a sketch of C. [2]
Find the area of the region enclosed by C, leaving your answer in terms of π. [7]
3](https://image.slidesharecdn.com/assignment1-201021064334/85/Assignment-1-polar-equation-revision-exercise-3-320.jpg)
![The curve C has polar equation r = 3 + 2 cos θ, for −π < θ ≤ π. The straight line l has polar equation
r cos θ = 2. Sketch both C and l on a single diagram. [3]
Find the polar coordinates of the points of intersection of C and l. [4]
The region R is enclosed by C and l, and contains the pole. Find the area of R. [6]
4](https://image.slidesharecdn.com/assignment1-201021064334/85/Assignment-1-polar-equation-revision-exercise-4-320.jpg)
![The curve C has polar equation r = 2 cos 2θ. Sketch the curve for 0 ≤ θ < 2π. [4]
Find the exact area of one loop of the curve. [4]
5](https://image.slidesharecdn.com/assignment1-201021064334/85/Assignment-1-polar-equation-revision-exercise-5-320.jpg)
![The curves C1 and C2 have polar equations given by
C1 : r = 3 sin θ, 0 ≤ θ < π,
C2 : r = 1 + sin θ, −π < θ ≤ π.
(i) Find the polar coordinates of the points, other than the pole, where C1 and C2 meet. [2]
(ii) In a single diagram, draw sketch graphs of C1 and C2. [3]
(iii) Show that the area of the region which is inside C1 but outside C2 is π. [5]
6](https://image.slidesharecdn.com/assignment1-201021064334/85/Assignment-1-polar-equation-revision-exercise-6-320.jpg)
![The curves C1
and C2
have polar equations
r = 4 cos θ and r = 1 + cos θ
respectively, where −1
2
π ≤ θ ≤ 1
2
π.
(i) Show that C1
and C2
meet at the points A 4
3
, α and B 4
3
, −α , where α is the acute angle such
that cos α = 1
3
. [2]
(ii) In a single diagram, draw sketch graphs of C1
and C2
. [3]
(iii) Show that the area of the region bounded by the arcs OA and OB of C1
, and the arc AB of C2
, is
4π − 1
3
√
2 − 13
2
α. [7]
7](https://image.slidesharecdn.com/assignment1-201021064334/85/Assignment-1-polar-equation-revision-exercise-7-320.jpg)
![The curve C has polar equation r = 2 sin 1 − cos , for 0 ≤ ≤ . Find
dr
d
and hence find the polar
coordinates of the point of C that is furthest from the pole. [5]
Sketch C. [2]
Find the exact area of the sector from = 0 to = 1
4
. [6]
8](https://image.slidesharecdn.com/assignment1-201021064334/85/Assignment-1-polar-equation-revision-exercise-8-320.jpg)
Assignment 1.polar equation revision exercise
- 1. 1 Find the cartesian equation corresponding to the polar equation r = ( √ 2) sec(θ − 1 4 π). [3] Sketch the the graph of r = ( √ 2) sec(θ − 1 4 π), for −1 4 π < θ < 3 4 π, indicating clearly the polar coordinates of the intersection with the initial line. [2] Assignment 1 Name:____________________ Group: ___________ Due: Thur, Sep 3rdPolar Coordinates Notice: Question 4 and 8 wil be marked.
- 2. The curve C has cartesian equation (x2 + y2 ) 2 = a2 (x2 − y2 ), where a is a positive constant. Show that C has polar equation r2 = a2 cos 2θ. [2] Sketch C for −π < θ ≤ π. [2] Find the area of the sector between θ = −1 4 π and θ = 1 4 π. [3] Find the polar coordinates of all points of C where the tangent is parallel to the initial line. [7] 2
- 3. The curve C has polar equation r = θ sin θ, where 0 ≤ θ ≤ π. Draw a sketch of C. [2] Find the area of the region enclosed by C, leaving your answer in terms of π. [7] 3
- 4. The curve C has polar equation r = 3 + 2 cos θ, for −π < θ ≤ π. The straight line l has polar equation r cos θ = 2. Sketch both C and l on a single diagram. [3] Find the polar coordinates of the points of intersection of C and l. [4] The region R is enclosed by C and l, and contains the pole. Find the area of R. [6] 4
- 5. The curve C has polar equation r = 2 cos 2θ. Sketch the curve for 0 ≤ θ < 2π. [4] Find the exact area of one loop of the curve. [4] 5
- 6. The curves C1 and C2 have polar equations given by C1 : r = 3 sin θ, 0 ≤ θ < π, C2 : r = 1 + sin θ, −π < θ ≤ π. (i) Find the polar coordinates of the points, other than the pole, where C1 and C2 meet. [2] (ii) In a single diagram, draw sketch graphs of C1 and C2. [3] (iii) Show that the area of the region which is inside C1 but outside C2 is π. [5] 6
- 7. The curves C1 and C2 have polar equations r = 4 cos θ and r = 1 + cos θ respectively, where −1 2 π ≤ θ ≤ 1 2 π. (i) Show that C1 and C2 meet at the points A 4 3 , α and B 4 3 , −α , where α is the acute angle such that cos α = 1 3 . [2] (ii) In a single diagram, draw sketch graphs of C1 and C2 . [3] (iii) Show that the area of the region bounded by the arcs OA and OB of C1 , and the arc AB of C2 , is 4π − 1 3 √ 2 − 13 2 α. [7] 7
- 8. The curve C has polar equation r = 2 sin 1 − cos , for 0 ≤ ≤ . Find dr d and hence find the polar coordinates of the point of C that is furthest from the pole. [5] Sketch C. [2] Find the exact area of the sector from = 0 to = 1 4 . [6] 8