Parametric Equations (A Level only) (OCR A Level Maths: Pure)

The curve C has parametric equations

  and

Complete the table below.

Plot the graph of C on the axes below.

A company’s logo is created using an arc of a circle as shown in the diagram below.

When the end points of the arc are joined to the origin, they form the major sector of a circle with angle  radians at the centre.

The arc is formed using the parametric equations

with domain 

Find the size of the angle , as well as the length of the arc used in the logo.

A curve C has parametric equations

Find a Cartesian equation for the curve C in the form .

A sketch of the graph with parametric equations

is shown below.

Find the value of t whenand hence write down the coordinates of the x-axis intercept.

Find the values of t whenand hence write down the coordinates of the y-axis intercepts.

Find the coordinates of the points where the graph intersects the line with equation .

A curve C is defined by the parametric equations

Show that the equation of the curve can be given in the Cartesian form ,  with  .

Assuming there are no restrictions on the values of t, find the domain and range of .

Sketch the graph of .

The curve defined parametrically by

where p is a non-zero constant, passes through the point (3 , -3).

Show that at point ,.

Hence, or otherwise, find the value of p.

A curve C is defined by the parametric equations

Show that.

Determine the ranges of x and y in the given domain of t.

Sketch the graph of C.

The curve C has parametric equations

  and  

A company’s logo is created using an arc of a circle and a straight line as shown in the diagram below.

The arc part of the logo is formed using the parametric equations

with domain  -

Find the equation of the line passing through points A and B, stating values to three significant figures where appropriate.

Find a Cartesian equation for the curve that has parametric equations

giving your answer in the form

A curve is defined by the parametric equations

By changing these equations into Cartesian form, show that this is the equation of a circle, and determine its centre and radius.

A sketch of the graph with parametric equations

is shown below.

Find the coordinates of all points where the graph intersects the coordinate axes.

Find the coordinates of the points where the graph intersects the line with equation .

A curve C is defined by the parametric equations

Show that the Cartesian equation of can be written in the form , where is a function of y.

Sketch the curve C.

The curve defined parametrically by

where is a constant, passes through the point (139 , 43).

Find the value of .

A curve C is defined by the parametric equations

Show that .

Sketch the graph of C.

The curve C has parametric equations

      and    

A company’s logo is a major arc of an ellipse as shown in the diagram below.

When the end points of the arc are connected to the origin, they form an elliptic sector with angle  radians at the centre.

The logo is formed using the parametric equations

with domain

Find the angle , giving your answer to three significant figures.

Find a Cartesian equation for the curve that has parametric equations

giving your answer in the form .

A curve is defined by the parametric equations

where a and b are non-zero constants, and p and q are constants.

Show that if then this is the equation of a circle, giving the equation of the circle in Cartesian form.

In the case where  and  , the curve is used to represent the position of a robot at time t.

Find the total distance travelled by the robot between times and .

A sketch of the graph with parametric equations

is shown below.

Find the coordinates of the points where the graph crosses the coordinate axes.

Find the coordinates of the points where the graph intersects the square with vertices at (-1 , 1), (1 , 1), (1 , -1), (-1 , -1).

A curve C has parametric equations

Determine the ranges of x and y in the given domain of t.

Show that the Cartesian equation of C can be written in the form 

Sketch the graph of  C.

The curve defined parametrically by

where is a constant, passes through the point (-22 , 0).

Find the possible values of .

Show that can be written as .

A curve C is defined by the parametric equations

Sketch the graph of C, showing clearly the beginning and end points of the curve, as well as the points of intersection with the coordinate axes and any minimum and maximum points.