Polar coordinates

The curve C has polar equation r = 2 + tanθ The curve C meets the line θ = at the point A The point B has polar coordinates (4, 0)

The diagram shows part of the curve C, and the points A and B

Find the area of the shaded region.

Give your answer in an exact form.

Show that r takes values in the range r k, where k is an integer.

Find the Cartesian equation of C in the form y² f (x)

The ellipse E has equation

Find the transformation that maps the graph of E onto C

Roberto is solving this mathematics problem:

Roberto’s solution is as follows:

Sketch the curve C₁

Explain what Roberto has done wrong.

Find the area enclosed by C₁

P and Q are distinct points on C₁ for which r is a maximum. P is above the initial line.

Find the polar coordinates of P and Q

T maps C₁ onto a curve C₂

T maps P onto the point P

Find the polar coordinates of P

Find the area enclosed by C₂

Fully justify your answer.

The curve C has polar equation

Find a Cartesian equation for C.

Circle your answer.

The line L has polar equation

Show that L is perpendicular to the initial line.

The curve C has polar equation

Find the polar coordinates of the points of intersection of L and C

Fully justify your answer.

The region R is the set of points such that

and

Find the exact area of R

The diagram shows part of a spiral curve.

The point P has polar coordinates (r, ) where 0

The points T and S lie on the initial line and O is the pole.

TPQ is the tangent to the curve at P.

Show that the gradient of TPQ is equal to

The curve has polar equation

where b is a constant such that 0 b

Use the result of part (a) to show that the angle between the line OP and the tangent TPQ does not depend on .