Coordinate Systems (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

The rectangular hyperbola has equation

Verify that, for the point is a general point on .

The point on has parameter

Show that the normal to at the point has equation

This normal at meets again at the point .

Find the coordinates of .

The rectangular hyperbola has equation where is a positive constant.

The point , where , lies on

Use calculus to show that an equation of the normal to at is

The parabola has equation

The normal to at the point with coordinates meets at the point where

Determine the exact coordinates of

Given that

• the point is the focus of

• the line is the directrix of

• the line through and meets at the point

determine the exact length of

The hyperbola has equation where is a positive constant.

The point , where , lies on .

The tangent to at meets the -axis at the point and meets the -axis at the point .

Determine, in terms of and ,

(i) the coordinates of ,

(ii) the coordinates of .

Given that the area of triangle , where is the origin, is 90 square units,

determine the value of , giving your answer as a simplified surd.

The parabola has equation

The point , where ,lies on .

Use calculus to show that the normal to at has equation

The normal to at the point where meets again at the point .

Determine the exact coordinates of .

The rectangular hyperbola has Cartesian equation .

The point with coordinates , where , lies on .

Use calculus to determine an equation for the normal to at the point .

Give your answer in the form .

Given that

determine the coordinates of the point where the normal meets again.

Give your answer in simplest form.

The point lies on the parabola with equation

Show that the point , where , lies on the parabola.

Show that the chord passes through the focus of the parabola.

The tangent to the parabola at and the tangent to the parabola at meet at the point

Determine, in simplest form, the coordinates of Determine

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The rectangular hyperbola has equation

The point , , where is a constant, is a general point on

State the value of

Show that the normal to at the point has equation

The points and lie on

The point has parameter and the point has parameter , where is a constant.

The normal to at meets again at

Determine the possible values of

A parabola has equation where is a positive constant.

The point is the focus of

The line with equation where is a positive constant, intersects at the point

Show that

The line passes through and intersects the directrix of on the -axis.

The line intersects the -axis at the point

Show that the coordinate of is

The line intersects the directrix of at the point

Given that the areas of triangles and , where is the origin, satisfy the ratio

area area

determine the exact value of

The parabola has equation

The point , where , lies on

Use calculus to show that the normal to at has equation

Hence find the equations of the two normals to which pass through the point (54, 0), giving your answers in the form where and are constants to be determined.

Given that

• the normals found in part (b) intersect the directrix of at the points and

• the point is the focus of

determine the area of triangle

The parabola has equation

The point is the focus of

Write down the coordinates of

Determine the exact perimeter of the triangle , where is the origin.

Give your answer in simplest form.

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The rectangular hyperbola has equation

The point lies on

Show, using calculus, that the normal to at has equation

The normal to at crosses again at the point

Determine an equation for the tangent to at , giving your answer in the form where and are rational constants.

The curve has equation

where is a positive constant.

The line with equation , where is a positive constant, intersects at the point .

Use calculus to determine, in terms of and , an equation for the tangent to at .

The tangent to at meets the -axis at the point and meets the -axis at the point .

Determine the coordinates of and the coordinates of , giving your answers in terms of and .

Hence show that the area of triangle , where is the origin, is independent of .

The parabola has equation

The point on has coordinates where is a non-zero constant.

Use calculus to show that the tangent to at has equation

The tangent to at meets the ‑axis at the point .

Write down the coordinates of .

The point is the focus of .

Write down the coordinates of .

The straight line passes through and .

The straight line passes through and , where is the origin.

Given that and intersect at the point ,

show that the coordinates of satisfy the equation

A rectangular hyperbola has equation

The point , , is a general point on .

Show that the equation of the tangent to at is

The distinct points and lie on . The tangent to at the point and the tangent to at the point meet at the point .

Find the coordinates of the points and .

The parabola has Cartesian equation

The point and the point , where , , are points on .

Show that an equation of the normal to at is

Write down an equation of the normal to at

The normal to at and the normal to at meet at the point

Show that has coordinates

The line , where is the origin, is perpendicular to the line .

Find the value of

The hyperbola has Cartesian equation

The parabola has parametric equations

The hyperbola intersects the parabola at the point .

Use algebra to determine the coordinates of .

The point with coordinates lies on .

Find an equation for the normal to at .

Give your answer in the form , where , and are integers to be determined.

Use algebra to determine, in simplest form, the exact coordinates of the points where this normal intersects the hyperbola .

The rectangular hyperbola has equation

The point , where , lies on .

Use calculus to show that the normal to at has equation

The normal to at meets again at the point

Determine, in terms of , the coordinates of , giving your answers in simplest form.

The parabola has equation , where is a positive constant.

The line with equation is a tangent to at the point .

Show that

Hence determine the coordinates of .

Given that the point is the focus of and that the line crosses the directrix of at the point ,

determine the exact area of triangle .