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Tangents & Normals (Edexcel International A Level (IAL) Further Maths): Revision Note

Exam code: YFM01

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

Tangents & Normals to a Parabola

What is a general point on a parabola?

  • open parentheses 3 comma space 6 close parentheses is a fixed point on the parabola y squared equals 12 x

  • open parentheses 3 t squared comma space 6 t close parentheses is a general point on the parabola y squared equals 12 x

    • It is algebraic

    • but it still satisfies the equation

    • and it can move along the curve (depending on t)

  • General points are formed from parametric equations

    • A general point on y squared equals 4 a x is open parentheses a t squared comma space 2 a t close parentheses

  • You can have two or more distinct general points

    • open parentheses 3 p squared comma space 6 p close parenthesesand open parentheses 3 q squared comma space 6 q close parentheses are general points P and Q on y squared equals 12 x

How do I find the tangent or normal to a parabola at a general point?

  • To find the tangent to the parabola y squared equals 12 x at the point open parentheses 3 t squared comma space 6 t close parentheses

    • use y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses

    • substitute in x subscript 1 equals 3 t squared and y subscript 1 equals 6 t

    • find m using calculus from fraction numerator straight d y over denominator straight d x end fraction at open parentheses 3 t squared comma space 6 t close parentheses

      • Make y the subject of y squared equals 12 x

      • y equals square root of 12 x space end root equals 2 square root of 3 space x to the power of 1 half end exponent

      • Ignore ±√ for now

      • fraction numerator straight d y over denominator straight d x end fraction equals 2 square root of 3 cross times 1 half x to the power of negative 1 half end exponent equals square root of 3 over x end root

      • At x subscript 1 equals 3 t squared, fraction numerator straight d y over denominator straight d x end fraction equals square root of fraction numerator 3 over denominator 3 t squared end fraction end root equals 1 over t

    • The tangent is y minus 6 t equals 1 over t open parentheses x minus 3 t squared close parentheses

  • The normal has a perpendicular gradient negative 1 over m

    • y minus 6 t equals negative t open parentheses x minus 3 t squared close parentheses

How do I use tangents and normals that pass through general points?

  • All methods in coordinate geometry still work

    • However points of intersection will be algebraic in t

  • Questions can involve forming and solving equations in t

Examiner Tips and Tricks

If you know parametric and implicit differentiation, they are also acceptable methods to find fraction numerator straight d y over denominator straight d x end fraction.

Worked Example

The point P with coordinates open parentheses 4 t squared comma space 8 t close parentheses lies on the parabola y squared equals 16 x.

(a) Use calculus to show that the equation of the normal at P is  y plus t x equals 4 t open parentheses t squared plus 2 close parentheses

The normal at P has the form y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses  
Substitute in x subscript 1 equals 4 t squared and y subscript 1 equals 8 t

y minus 8 t equals m open parentheses x minus 4 t squared close parentheses 

The gradient of the normal is the negative reciprocal of the gradient of the tangent
Find the gradient of the tangent
For example first make y  the subject of the curve
(Use the positive square root)

y equals square root of 16 x end root equals 4 x to the power of 1 half end exponent 

Then differentiate

table row cell fraction numerator straight d y over denominator straight d x end fraction end cell equals cell 4 cross times 1 half x to the power of negative 1 half end exponent end cell row blank equals cell fraction numerator 2 over denominator square root of x end fraction end cell end table
 

Then substitute in x subscript 1 equals 4 t squared

table row cell fraction numerator straight d y over denominator straight d x end fraction end cell equals cell fraction numerator 2 over denominator square root of 4 t squared end root end fraction end cell row blank equals cell 1 over t end cell end table

The negative reciprocal is negative t
Therefore the equation of the normal is

y minus 8 t equals negative t open parentheses x minus 4 t squared close parentheses 

Expand and rearrange

table row cell y minus 8 t end cell equals cell negative t x plus 4 t cubed end cell row y equals cell negative t x plus 4 t cubed plus 8 t end cell row cell y plus t x end cell equals cell 4 t cubed plus 8 t end cell end table

Then factorise the right-hand side

y plus t x equals 4 t open parentheses t squared plus 2 close parentheses

(b) Hence find the coordinates of the three points on the parabola at which the normal passes through the point open parentheses 9 comma space 0 close parentheses.

The equation of the normal above must pass through the point open parentheses 9 comma space 0 close parentheses
Substitute in x equals 9 and y equals 0

0 plus 9 t equals 4 t open parentheses t squared plus 2 close parentheses 

This forms an equation in t
Rearrange and solve

table row cell 9 t end cell equals cell 4 t cubed plus 8 t end cell row 0 equals cell 4 t cubed minus t end cell row 0 equals cell t open parentheses 4 t squared minus 1 close parentheses end cell row 0 equals cell t open parentheses 2 t minus 1 close parentheses open parentheses 2 t plus 1 close parentheses end cell end table 

t equals 0 comma space space t equals 1 half comma space space t equals negative 1 half

The question asks for the coordinates of the points on the curve, P, open parentheses 4 t squared comma space 8 t close parentheses
Substitute in the three values of t

open parentheses 0 comma space 0 close parentheses comma space space open parentheses 1 comma space 4 close parentheses comma space space open parentheses 1 comma space minus 4 close parentheses

It helps to imagine the situation visually (below) to check the answers make sense

Normals to a general parabola

Tangents & Normals to a Rectangular Hyperbola

What is a general point on a rectangular hyperbola?

  • open parentheses 5 comma space 5 close parentheses is a fixed point on the rectangular hyperbola x y equals 25

  • open parentheses 5 t comma space fraction numerator space 5 over denominator t end fraction close parentheses is a general point on the rectangular hyperbola x y equals 25

    • It is algebraic

    • but it still satisfies the equation

    • and it can move along the curve (depending on t)

      • t not equal to 0

  • General points are formed from parametric equations

    • A general point on x y equals c squared is open parentheses c t comma space c over t close parentheses

  • You can have two or more distinct general points

    • open parentheses 5 p comma space 5 over p close parentheses and open parentheses 5 q comma space 5 over q close parentheses are general points P and Q on x y equals 25

How do I find the tangent or normal to a rectangular hyperbola at a general point?

  • To find the tangent to the rectangular hyperbola x y equals 25 at the point open parentheses 5 t comma space 5 over t close parentheses

    • use y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses

    • substitute in x subscript 1 equals 5 t and y subscript 1 equals 5 over t

    • find m using calculus from fraction numerator straight d y over denominator straight d x end fraction at open parentheses 5 t comma space 5 over t close parentheses

      • Make y the subject of x y equals 25

      • y equals 25 over x equals 25 x to the power of negative 1 end exponent

      • fraction numerator straight d y over denominator straight d x end fraction equals 25 cross times open parentheses negative 1 close parentheses x to the power of negative 2 end exponent equals negative 25 over x squared

      • At x subscript 1 equals 5 t, fraction numerator straight d y over denominator straight d x end fraction equals negative 25 over open parentheses 5 t close parentheses squared equals negative 1 over t squared

    • The tangent is y minus 5 over t equals negative 1 over t squared open parentheses x minus 5 t close parentheses

  • The normal has a perpendicular gradient negative 1 over m

    • y minus 5 over t equals t squared open parentheses x minus 5 t close parentheses

How do I use tangents and normals that pass through general points?

  • All methods in coordinate geometry still work

    • However points of intersection will be algebraic in t

  • Questions can involve forming and solving equations in t

Examiner Tips and Tricks

If you know parametric and implicit differentiation, they are also acceptable methods to find fraction numerator straight d y over denominator straight d x end fraction.

Worked Example

The point P with coordinates open parentheses 2 p comma space 2 over p close parentheses and the point Q with coordinates open parentheses 2 q comma space 2 over q close parentheses lie on the rectangular hyperbola x y equals 4, where p not equal to plus-or-minus q.

(a)  Use calculus to show that the equation of the tangent to the curve at P is  p squared y equals negative x plus 4 p.

The tangent at P  has the form y minus y subscript 1 equals m open parentheses x minus x subscript 1 close parentheses  
Substitute in x subscript 1 equals 2 p and y subscript 1 equals 2 over p

y minus 2 over p equals m open parentheses x minus 2 p close parentheses

Find the gradient of the tangent, for example, first make y  the subject of the curve

y equals 4 over x equals 4 x to the power of negative 1 end exponent

Then differentiate

table row cell fraction numerator straight d y over denominator straight d x end fraction end cell equals cell negative 4 x to the power of negative 2 end exponent end cell row blank equals cell negative 4 over x squared end cell end table

Then substitute in x subscript 1 equals 2 p and simplify

table row cell fraction numerator straight d y over denominator straight d x end fraction end cell equals cell negative 4 over open parentheses 2 p close parentheses squared end cell row blank equals cell negative fraction numerator 4 over denominator 4 p squared end fraction end cell row blank equals cell negative 1 over p squared end cell end table

Therefore the equation of the tangent is

y minus 2 over p equals negative 1 over p squared open parentheses x minus 2 p close parentheses

Multiply both sides by p squared then rearrange

table row cell p squared y minus 2 p end cell equals cell negative open parentheses x minus 2 p close parentheses end cell row cell p squared y minus 2 p end cell equals cell negative x plus 2 p end cell end table

Make p squared y the subject

p squared y equals negative x plus 4 p

(b)  Hence find the x-coordinate of the point of intersection between the tangent at P and the tangent at Q, giving your answer in fully simplified form.

Finding the tangent at Q  requires the exact same steps as above, but with q  instead of p
You can therefore write down the tangent at Q  with no working

q squared y equals negative x plus 4 q

The tangent at P  intersects the tangent at Q 

Tangents to a rectangular hyperbola intersecting each other

Make y the subject of each tangent and set them equal to each other

fraction numerator negative x plus 4 p over denominator p squared end fraction equals fraction numerator negative x plus 4 q over denominator q squared end fraction

Cross-multiply and expand

table row cell q squared open parentheses negative x plus 4 p close parentheses end cell equals cell p squared open parentheses negative x plus 4 q close parentheses end cell row cell negative q squared x plus 4 p q squared end cell equals cell negative p squared x plus 4 p squared q end cell end table 

Bring the x  terms to one side and factorise

p squared x minus q squared x equals 4 p squared q minus 4 p q squared open parentheses p squared minus q squared close parentheses x equals 4 p q open parentheses p minus q close parentheses

Make x  the subject and use the difference of two squares

table row x equals cell fraction numerator 4 p q open parentheses p minus q close parentheses over denominator p squared minus q squared end fraction end cell row x equals cell fraction numerator 4 p q open parentheses p minus q close parentheses over denominator open parentheses p minus q close parentheses open parentheses p plus q close parentheses end fraction end cell end table

As p not equal to plus-or-minus q, the brackets will never be zero
This means the open parentheses p minus q close parentheses bracket can be cancelled, giving the final answer

x equals fraction numerator 4 p q over denominator p plus q end fraction

Note: if p equals q then point P  is the same as point Q
Also, if p equals negative q, then the tangents at P  and Q  are parallel

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    Author
    Reviewer
    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.