A LevelFurther MathsEdexcelFurther Pure 1Revision NotesCoordinate SystemsLoci ProblemsLoci Problems

Loci Problems (Edexcel A Level Further Maths: Further Pure 1): Revision Note

Exam code: 9FM0

Mark Curtis

Written by: Mark Curtis

Updated on

Loci problems

What is a locus?

  • A locus is a set of points

    • that is traced out

    • from a set of instructions

Diagram of an ellipse centred at O with major axes labelled from -a to a and minor axis from -b to b. Points P, Q, R are marked, where the tangent at P is perpendicular to the tangent at Q and the tangents intersect at R which lies on a dotted circle.

  • Here is an example of a set of instructions

    • Find any point P on an ellipse, x squared over a squared plus y squared over b squared equals 1

      • Draw its tangent

    • Find the corresponding point Q on the ellipse

      • whose tangent is perpendicular

    • Find the point of intersection, R, of the two tangents

    • Repeat this process for different starting points, P

    • As you vary P

      • Q will vary

      • R will vary

      • R will trace out a circle

    • i.e. the locus of R is a circle

  • The equation of the locus can be found using algebra

    • giving x squared plus y squared equals a squared plus b squared

Examiner Tips and Tricks

Loci questions often carry a lot of marks in the exam as they involve multiple steps, problem solving and complicated algebra!

What are some common loci problems?

  • Some common loci problems include

    • finding the point of intersection of two tangents / normals

    • enforcing specific conditions on tangents and normals

      • e.g. they must pass through the origin

    • finding midpoints

      • open parentheses fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction comma space fraction numerator y subscript 1 plus y subscript 2 over denominator 2 end fraction close parentheses

    • finding points of re-intersection with the curve

      • e.g. a normal at P cuts the curve again at Q

Examiner Tips and Tricks

Exam questions don't always use the word 'locus', so you may need to spot the clues yourself (e.g. wording like "as P varies..." or "show that the points lie on a curve").

How do I find the equation of a locus?

  • To find the equation of a locus:

    • Follow the instructions in the question to find a set of coordinates in terms of a parameter, t or theta

      • e.g. a midpoint given by open parentheses straight f open parentheses t close parentheses comma space straight g open parentheses t close parentheses close parentheses

    • Set the coordinate components equal to new variables X and Y

      • X equals straight f open parentheses t close parentheses and Y equals straight g open parentheses t close parentheses

      • These are parametric equations of the locus

      • They also show the range of values X and Y can take

    • Eliminate the parameter

      • This gives the Cartesian equation of the locus in terms of X and Y

      • This may involve trig or hyperbolic identities

    • Present the equation of the locus in lower case letters at the end

      • i.e. x and y

      • even though the working is in X and Y

Examiner Tips and Tricks

Capital letters X and Y help avoid confusion when performing calculations with x and y from tangent or curve equations.

Example with intersecting tangents

Worked Example

The tangent to the rectangular hyperbola x y equals c squared at the point open parentheses c t comma space c over t close parentheses is given by the equation

x plus t squared y equals 2 c t

The tangents at two distinct points on the curve, P open parentheses c p comma space c over p close parentheses and Q open parentheses c q comma space c over q close parentheses, intersect at the point R, as shown.

Graph showing the hyperbola xy = c^2 with points P(cp, c/p) and Q(cq, c/q) on the curve in the first quadrant. The tangent at P and the tangent at Q intersect at the point R.

(a) Find the coordinates of the point R in terms of p and q.

(b) Given that p q equals 10, find the locus of R as P varies.

Answer:

(a)

Find the equations of the tangents at P and Q

  • by substituting t equals p and t equals q into the tangent equation given in the question

x plus p squared y equals 2 c p x plus q squared y equals 2 c q

Find the point of intersection of the two tangents

  • by solving the equations simultaneously to find x and y

For example, subtract the two equations to eliminate x and find y

table row cell p squared y minus q squared y end cell equals cell 2 c p minus 2 c q end cell row cell open parentheses p squared minus q squared close parentheses y end cell equals cell 2 c open parentheses p minus q close parentheses end cell row y equals cell fraction numerator 2 c open parentheses p minus q close parentheses over denominator p squared minus q squared end fraction end cell end table

Use the difference of two squares and that fact that P and Q are distinct (p not equal to q) to simplify the y-coordinate

table row y equals cell fraction numerator 2 c up diagonal strike open parentheses p minus q close parentheses end strike over denominator open parentheses p plus q close parentheses up diagonal strike open parentheses p minus q close parentheses end strike end fraction end cell row y equals cell fraction numerator 2 c over denominator p plus q end fraction end cell end table

Find the x-coordinate

  • e.g. by substituting y into the first tangent equation

table row cell x plus p squared open parentheses fraction numerator 2 c over denominator p plus q end fraction close parentheses end cell equals cell 2 c p end cell row x equals cell 2 c p minus p squared open parentheses fraction numerator 2 c over denominator p plus q end fraction close parentheses end cell end table

Simplify the expression by adding algebraic fractions

table row x equals cell 2 c p minus fraction numerator 2 c p squared over denominator p plus q end fraction end cell row x equals cell fraction numerator 2 c p open parentheses p plus q close parentheses minus 2 c p squared over denominator p plus q end fraction end cell row x equals cell fraction numerator up diagonal strike 2 c p squared end strike plus 2 c p q up diagonal strike negative 2 c p squared end strike over denominator p plus q end fraction end cell row x equals cell fraction numerator 2 c p q over denominator p plus q end fraction end cell end table

Present your answer as coordinates

R open parentheses fraction numerator 2 c p q over denominator p plus q end fraction comma space fraction numerator 2 c over denominator p plus q end fraction close parentheses

(b)

Substitute p q equals 10 into the x-coordinate

table row blank blank cell R open parentheses fraction numerator 20 c over denominator p plus q end fraction comma space fraction numerator 2 c over denominator p plus q end fraction close parentheses end cell end table

Let X and Y be parametric equations of the locus of R

  • Set them equal to the x and y coordinates of R

table row X equals cell fraction numerator 20 c over denominator p plus q end fraction end cell row Y equals cell fraction numerator 2 c over denominator p plus q end fraction end cell end table

Eliminate the parameters p and q

  • A quick way here is to write Y in terms of X

  • or divide Y by X

table row Y equals cell fraction numerator 2 c over denominator p plus q end fraction end cell row Y equals cell 1 over 10 open parentheses fraction numerator 20 c over denominator p plus q end fraction close parentheses end cell row Y equals cell 1 over 10 X end cell end table

This is the Cartesian equation of the locus (a straight line through the origin)

Present the equation of the locus in lower case letters

The locus of R is the straight line y equals 1 over 10 x

Example with trigonometric identities

Worked Example

The normal to the ellipse x squared over a squared plus y squared over b squared equals 1 at P open parentheses a cos theta comma space b sin theta close parentheses has the equation

open parentheses a sin theta close parentheses x minus open parentheses b cos theta close parentheses y equals open parentheses a squared minus b squared close parentheses sin theta cos theta

The normal at P intersects the x-axis at the point C.

The point M is the midpoint of the line segment C P, as shown.

Diagram of an ellipse with major axis labelled -a to a and minor axis -b to b. Point P on ellipse with coordinates (a cos θ, b sin θ) and the normal to P shown intersecting the x-axis at C. The midpoint M of PC is marked.

Show that, as theta varies from 0 to 2 pi, the point M lies on a different ellipse, whose equation you should find.

Answer:

The wording suggests that this is a locus question

Find the coordinates of C by setting y equals 0 in the equation of the normal

table row cell open parentheses a sin theta close parentheses x minus open parentheses b cos theta close parentheses cross times 0 end cell equals cell open parentheses a squared minus b squared close parentheses sin theta cos theta end cell row cell open parentheses a sin theta close parentheses x end cell equals cell open parentheses a squared minus b squared close parentheses sin theta cos theta end cell row x equals cell fraction numerator a squared minus b squared over denominator a end fraction cos theta end cell end table

It helps to write out the coordinates of C

C open parentheses fraction numerator a squared minus b squared over denominator a end fraction cos theta comma space 0 close parentheses

Use open parentheses fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction comma space fraction numerator y subscript 1 plus y subscript 2 over denominator 2 end fraction close parentheses to find M, the midpoint of C open parentheses fraction numerator a squared minus b squared over denominator a end fraction cos theta comma space 0 close parentheses and P open parentheses a cos theta comma space b sin theta close parentheses

M open parentheses fraction numerator fraction numerator a squared minus b squared over denominator a end fraction cos theta plus a cos theta over denominator 2 end fraction comma space fraction numerator 0 plus b sin theta over denominator 2 end fraction close parentheses

Simplify the coordinates (e.g. by adding the algebraic fractions)

equals M open parentheses fraction numerator open parentheses fraction numerator a squared minus b squared over denominator a end fraction plus a close parentheses cos theta over denominator 2 end fraction comma space fraction numerator b sin theta over denominator 2 end fraction close parentheses equals M open parentheses fraction numerator open parentheses fraction numerator a squared minus b squared over denominator a end fraction plus fraction numerator a squared over denominator a end fraction close parentheses cos theta over denominator 2 end fraction comma space fraction numerator b sin theta over denominator 2 end fraction close parentheses equals M open parentheses fraction numerator open parentheses fraction numerator 2 a squared minus b squared over denominator a end fraction close parentheses cos theta over denominator 2 end fraction comma space fraction numerator b sin theta over denominator 2 end fraction close parentheses equals M open parentheses fraction numerator open parentheses 2 a squared minus b squared close parentheses cos theta over denominator 2 a end fraction comma space fraction numerator b sin theta over denominator 2 end fraction close parentheses

Let X and Y be parametric equations of the locus of M

  • Set them equal to the x and y coordinates of M

table row X equals cell fraction numerator open parentheses 2 a squared minus b squared close parentheses cos theta over denominator 2 a end fraction end cell row Y equals cell fraction numerator b sin theta over denominator 2 end fraction end cell end table

Eliminate the parameter, by first making cos theta and sin theta the subjects of the equations

table row cell fraction numerator 2 a X over denominator 2 a squared minus b squared end fraction end cell equals cell cos theta end cell row cell fraction numerator 2 Y over denominator b end fraction end cell equals cell sin theta end cell end table

Then substituting these into the identity cos squared theta plus sin squared theta identical to 1

table row cell open parentheses fraction numerator 2 a X over denominator 2 a squared minus b squared end fraction close parentheses squared plus open parentheses fraction numerator 2 Y over denominator b end fraction close parentheses squared end cell equals 1 end table

This is almost in the form of an ellipse X squared over A squared plus Y squared over B squared equals 1

Rewrite the coefficients in the correct form

table row cell fraction numerator X squared over denominator space open parentheses fraction numerator 2 a squared minus b squared over denominator 2 a end fraction close parentheses squared end fraction plus Y squared over open parentheses b over 2 close parentheses squared end cell equals 1 end table

Present final equation in lower case letters

The locus of M is an ellipse with equation table row cell fraction numerator x squared over denominator space open parentheses fraction numerator 2 a squared minus b squared over denominator 2 a end fraction close parentheses squared end fraction plus y squared over open parentheses b over 2 close parentheses squared end cell equals 1 end table

Example with a chord intersecting the curve twice

Worked Example

The point P open parentheses 2 t squared comma space 4 t close parentheses lies on the parabola y squared equals 8 x.

The focus of the parabola is at the point F open parentheses 2 comma space 0 close parentheses.

The straight line through P and F intersects the curve at a second point, Q.

Let M be the midpoint of the chord P Q, as shown.

Graph of parabola y^2 = 8x with the point P with x-coordinate 2t^2 and y-coordinate 4t, shown on the graph. The focus F with coordinates (2, 0) is also shown on the x-axis, and the chord through P and F is shown, which intersects the curve again at the point Q.

Show that the locus of M is the parabola with equation y squared equals 4 x minus 8.

Answer:

You need to find the equation of the straight line through P and F, then find the point of intersection of this line with the curve y squared equals 8 x

Start by finding the gradient of P F and simplify

table row cell fraction numerator 4 t minus 0 over denominator 2 t squared minus 2 end fraction end cell equals cell fraction numerator 4 t over denominator 2 open parentheses t squared minus 1 close parentheses end fraction end cell row blank equals cell fraction numerator 2 t over denominator t squared minus 1 end fraction end cell end table

Substitute the gradient and the coordinates of F (or P) into y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses and simplify

table row cell y minus 0 end cell equals cell fraction numerator 2 t over denominator t squared minus 1 end fraction open parentheses x minus 2 close parentheses end cell row y equals cell fraction numerator 2 t open parentheses x minus 2 close parentheses over denominator t squared minus 1 end fraction end cell end table

Next, solve the simultaneous equations of this line with the curve y squared equals 8 x

  • e.g. substitute y from the line into the curve

open parentheses fraction numerator 2 t open parentheses x minus 2 close parentheses over denominator t squared minus 1 end fraction close parentheses squared equals 8 x

Rearrange the result into a three-term quadratic in x

table row cell fraction numerator 4 t squared open parentheses x minus 2 close parentheses squared over denominator open parentheses t squared minus 1 close parentheses squared end fraction end cell equals cell 8 x end cell row cell t squared open parentheses x minus 2 close parentheses squared end cell equals cell 2 x open parentheses t squared minus 1 close parentheses squared end cell row cell t squared open parentheses x squared minus 4 x plus 4 close parentheses end cell equals cell 2 x open parentheses t to the power of 4 minus 2 t squared plus 1 close parentheses end cell row cell t squared x squared up diagonal strike negative 4 t squared x end strike plus 4 t squared end cell equals cell 2 t to the power of 4 x up diagonal strike negative 4 t squared x end strike plus 2 x end cell row cell t squared x squared plus 4 t squared end cell equals cell 2 t to the power of 4 x plus 2 x end cell row cell t squared x squared minus 2 t to the power of 4 x minus 2 x plus 4 t squared end cell equals 0 row cell t squared x squared minus 2 open parentheses t to the power of 4 plus 1 close parentheses x plus 4 t squared end cell equals 0 end table

Factorise the quadratic, knowing that the x-coordinate of P open parentheses 2 t squared comma space 4 t close parentheses is already one of the solutions

  • i.e. open parentheses x minus 2 t squared close parentheses must be a factor

table row cell open parentheses x minus 2 t squared close parentheses open parentheses... close parentheses end cell equals 0 row cell open parentheses x minus 2 t squared close parentheses open parentheses t squared x... close parentheses end cell equals 0 row cell open parentheses x minus 2 t squared close parentheses open parentheses t squared x minus 2 close parentheses end cell equals 0 end table

Solve the equation to find two x-coordinates of intersection between the straight line and the curve

x equals 2 t squared space space and space space x equals 2 over t squared

The first x-coordinate is from P so the second must be from Q

Find the corresponding y-coordinate of Q

  • e.g. substitute x equals 2 over t squaredinto the straight line equation table row y equals cell fraction numerator 2 t open parentheses x minus 2 close parentheses over denominator t squared minus 1 end fraction end cell end table

    • then simplify

  • Avoid substituting into the curve equation y squared equals 8 x

    • as it gives two possibilities, y equals plus-or-minus..., making it harder

table row y equals cell fraction numerator 2 t open parentheses 2 over t squared minus 2 close parentheses over denominator t squared minus 1 end fraction end cell row y equals cell fraction numerator 2 t open parentheses fraction numerator 2 minus 2 t squared over denominator t squared end fraction close parentheses over denominator t squared minus 1 end fraction end cell row y equals cell fraction numerator negative 4 open parentheses fraction numerator t squared minus 1 over denominator t end fraction close parentheses over denominator t squared minus 1 end fraction end cell row y equals cell negative 4 open parentheses fraction numerator t squared minus 1 over denominator t end fraction close parentheses divided by open parentheses t squared minus 1 close parentheses end cell row y equals cell negative fraction numerator 4 up diagonal strike open parentheses t squared minus 1 close parentheses end strike over denominator t end fraction cross times fraction numerator 1 over denominator up diagonal strike open parentheses t squared minus 1 close parentheses end strike end fraction end cell row y equals cell negative 4 over t end cell end table

It helps to write out the coordinates of Q

Q open parentheses 2 over t squared comma space minus 4 over t close parentheses

Use open parentheses fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction comma space fraction numerator y subscript 1 plus y subscript 2 over denominator 2 end fraction close parentheses to find M, the midpoint of P open parentheses 2 t squared comma space 4 t close parentheses and Q open parentheses 2 over t squared comma space minus 4 over t close parentheses

M open parentheses fraction numerator 2 t squared plus 2 over t squared over denominator 2 end fraction comma space fraction numerator 4 t plus open parentheses negative 4 over t close parentheses over denominator 2 end fraction close parentheses equals M open parentheses t squared plus 1 over t squared comma space 2 t minus 2 over t close parentheses

Let X and Y be parametric equations of the locus of M

  • Set them equal to the x and y coordinates of M

X equals t squared plus 1 over t squared Y equals 2 t minus 2 over t

It is not immediately clear how to eliminate t

  • It helps to look at the answer given in the question

    • y squared equals 4 x minus 8

This suggests finding Y squared

table row cell Y squared end cell equals cell open parentheses 2 t minus 2 over t close parentheses squared end cell row blank equals cell 4 t squared minus fraction numerator 8 t over denominator t end fraction plus 4 over t squared end cell row blank equals cell 4 t squared minus 8 plus 4 over t squared end cell end table

Now that can be rewritten to show the relationship with X

table row cell Y squared end cell equals cell 4 t squared plus 4 over t squared minus 8 end cell row blank equals cell 4 open parentheses t squared plus 1 over t squared close parentheses minus 8 end cell row blank equals cell 4 X minus 8 end cell end table

Rewrite the final answer in lower case letters

The locus of M is a parabola with equation table row cell y squared end cell equals cell 4 x minus 8 end cell end table

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Mark Curtis

Author: Mark Curtis

Expertise: Maths Content Creator

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.