Differentiation of Parametric Equations (CIE A Level Maths: Pure 3)

Find an expression for  in terms of for the parametric equations

The graph of  y  against  x  passes through the point P (1 , 1).

A crane swings a wrecking ball along a two-dimensional path defined by the parametric equations 

as shown in the diagram below.

x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds.  Point A indicates the initial position of the wrecking ball, at time .

Find the height of the wrecking ball after 0.3 seconds.

Find the minimum height of the wrecking ball during its motion.

Find the horizontal distances from point A at the times when the wrecking ball is at a height of 2.9 m, giving your answers accurate to 1 decimal place.

The graph of the curve C shown below is defined by the parametric equations

Write down the value of    at the point (0 , 6).

Write down the value of  at the points (-3 , 3) and (3 , 3).

Find an expression for   in terms of .

The curve C has parametric equations

Find an expression, in terms of t, for  .

The curve C has parametric equations

Show that, in terms of , 

Show that the distance between the maximum and minimum points on C is  square units.

Find an expression for    in terms of t for the parametric equations

Verify that the graph of x against y passes through the point (0 , 1) and find the gradient at that point.

A crane swings a wrecking ball along a two-dimensional path defined by the parametric equations

as shown in the diagram below.

x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds.  Point A indicates the initial position of the wrecking ball, at time.

Find a Cartesian equation of the curve in the form    and state the domain of .

Find the difference between the maximum and minimum heights of the wrecking ball during its motion.

The crane is positioned such that point A is 7 m horizontally from the wall the wrecking ball is to destroy.
Find the height at which the wrecking ball will strike the wall.

The graph of the curve C shown below is defined by the parametric equations

Find an expression for    in terms of  .

The curve C has parametric equations

Find an expression, in terms of  t, for  .

The curve has parametric equations

Show that at the point (0 , 6),and find the value of    at this point.

The tangent at the point (0 , 6) is parallel to the normal at the point P.
Find the exact coordinates of point P

A curve C has parametric equations

The tangents to C at the points R and S meet at the point T, as shown in the diagram below.

Given that the x-coordinate of both points R and S is 5, find the area of the triangle RST.

A crane swings a wrecking ball along a two-dimensional path defined by the parametric equations

as shown in the diagram below.

x and y are, respectively, the horizontal and vertical displacements in metres from the origin, O, and t is the time in seconds.  Point A indicates the initial position of the wrecking ball.

The destruction of a building requires the wrecking ball to strike it at a height of 1.4 m whilst on the upward part of its path.
Find the horizontal distance from point A at which the ball hits the building.

The graph of the ellipse E shown below is defined by the parametric equations

Find an expression for    in terms of θ.

Find the equation of the tangent to E, at the point where  , giving your answer in the form , where a and b are real numbers that should be given in exact form.

The curve C has parametric equations

Find the equation of the normal to C at the point where C intersects the line  .

The graph of the curve defined by the parametric equations

is shown below.

Prove that the line with equation is not the normal to the curve at the point (1 , 1).

The diagram below shows a sketch of the curve defined by the parametric equations

The tangents to the curve that pass through the origin meet the curve at points A and B

Show that the values of t at points A  and B are    and  .

Hence, or otherwise, show that the area of the triangle OAB is  square units.