Y2, C5- Polar Coordinates

Question 1

What does a polar coordinate equation look like

Answer 1

P(r, θ)
Where r = length from pole
θ = angle anticlockwise from x axis measured in radians

Question 2

What is the Cartesian equation for r = 5

Answer 2

x^2 + y^2 = 25

Question 3

What equations do you need to know for x and y when converting polar coordinates into Cartesian equations

Answer 3

r^2 = x^2 + y^2
x = r * cos θ
y = r * sin θ
θ = arctan(y/x)

Question 4

Write r^2 = sin(θ + pi/4) in Cartesian form

Answer 4

r^2 = sinθcos(pi/4) + cosθsin(pi/4)
r^2 = (1/root(2))sinθ + (1/root(2))cosθ
root(2)r^3 = rsinθ + rcosθ
root(2)(x^2 + y^2)^3/2 = x + y

Question 4

What do polar equations usually start with (LHS)

Answer 4

r =
OR
r^2 =

Question 5

Convert x^2 - y^2 = 5 into a polar equation

Answer 5

r^2 * cos^2(θ) - r^2 * sin^2(θ) = 5
r^2 (cos^2(θ) - sin^2(θ)) = 5
r^2 (cos(2θ)) = 5
r^2 = 5sec(2θ)

Question 6

How would you sketch:
r = a
θ = alpha
r = aθ

Answer 6

1) Circle, radius a
2) Half line, angle alpha from +ve x axis
3) Spiral

Question 7

What does the edexcel spec remove in polar coordinates

Answer 7

Negative values for r

Question 8

How many +ve petals are there for the a polar equation in the form r^2 = a^2 * cos(3θ)

Answer 8

3 +ve petals
Coefficient before θ is the number of petals

Question 9

r = a(p + qcosθ), what shape do you get if p = q

Answer 9

You get a cardioid (egg with dimple) where the curve reaches the origin when θ = pi

Question 10

r = a(p + qcosθ), what shape do you get if p >= 2q

Answer 10

Egg / oval shape (if q = 0, circle centred at origin)

Question 11

r = a(p + qcosθ), what shape do you get if q < p < 2q

Answer 11

Dimple shape, unlike cardioid, centre will NEVER be at origin as r > 0 and not equal to 0

Question 12

Describe the shape of r = 2acosθ

Answer 12

Radius a, centre (a, 0)

Question 13

Describe the shape of r = a(1 - cosθ)

Answer 13

Backwards dimple

Question 14

Describe the shape of r = asec(θ)

Answer 14

Vertical straight line, x = a

Question 15

What is the formula for the area of each sector of a polar petal

Answer 15

0.5 * r^2 * dθ

Question 16

What is the integration formula polar integration

Answer 16

0.5 * int(r^2) dθ from beta to alpha

Question 17

How do you find the strategy of combines shaded areas

Answer 17

Draw a line from origin to the intersection to split the shaded area and then integrate under each curve separately

Question 18

When can you find tangents and normals to parametric curves

Answer 18

When they are parallel or perpendicular to the initial line
dx / dθ = 0
OR
dy / dθ = 0

Question 19

With polar coordinates, what is (dy/dθ) / (dx/dθ) equal to?

Answer 19

dy / dx

Question 20

When perpendicular to the initial line, what is dx/dθ equal to

Answer 20

0

Question 21

When parallel to the initial line, what is dy/dθ equal to

Answer 21

0

Question 22

Curve C has polar equation: r = 1 + 2cosθ, 0 < θ < pi/2
At point P on C, tangent to C is parallel to initial line. Find exact length of OP

Answer 22

y = rsinθ
y = 2sinθcosθ + sinθ
y = sin2θ + sinθ
dy / dθ = 4cos^(θ) + cosθ - 2 = 0
θ = (-1+root(33)) / 8
1 = 2cosθ = (3+root(33) / 4

Question 23

What is the proof for r = p + qcosθ has a dimple if p < 2q

Answer 23

There will be 3 tangents rather than 2 (which an egg has)
Find the 3 tangents
sinθ = 0
OR
cosθ = -p / 2q
If p > 2q, cosθ < -1, no solutions so not other tangent
If p = 2q, cosθ = -1 which gives θ = pi which is already a solution and so not another tangent