# Edexcel A Level Further Maths: Core Pure:复习笔记2.2.3 Invariant Points & Lines

### Invariant Points

#### What is an invariant point?

• When applying transformations to a shape or collection of points, there may be some points that stay in their original position; these are known as invariant points

#### How can I find invariant points?

• If the point given by position vector , is invariant under transformation  then we can say that
• This will create a system of simultaneous equations which can be solved to find the invariant point
• The origin (0,0) is always invariant under a linear transformation

#### Exam Tip

• Where the question allows, use your calculator to help solve the simultaneous equations
• Test your found invariant point by multiplying it by the transformation matrix, and making sure you still end up with the same point (invariant)

#### Worked Example

Find any invariant points under the transformation given by .

### A Line of Invariant Points

#### What is a line of invariant points?

• If every point on a line is mapped to itself under a particular transformation, then it is a line of invariant points
• For example, a line of reflection is a line of invariant points

#### How can I find a line of invariant points?

• Use the same strategy as for finding a single invariant point:
• If the point given by position vector , is invariant under transformation then we can say that
• This will create a system of simultaneous equations which can be solved to find the invariant point(s)
• If there is a line of invariant points, rather than solving to find a single solution (a point), the two equations will be able to simplify to the same equation
• This means that there are infinitely many solutions, and therefore infinitely many invariant points
• A line contains infinitely many points
• Your solution will be the equation of the invariant line e.g. y=3x

#### Exam Tip

• It may not always be obvious that the two equations reduce to the same thing (they could be an awkward multiple of each other)
• Use your calculator’s simultaneous equation solver; it will tell you that there are infinitely many solutions

#### Worked Example

Find the equation of the line of invariant points under the transformation given by

### Invariant Lines

#### What’s the difference between a line of invariant points and an invariant line?

• If every point on a line is mapped to itself under a particular transformation, then it is a line of invariant points
• Every single point on the line must stay in the same place
• With an invariant line however, every point on the line must simply map to another point on the same line
• We are only concerned with the overall line; not the individual points

#### How do I find an invariant line?

• We can use a similar strategy to finding invariant points, with two slight changes
• Use  to write the original position vector as
• Write the transformed position vector as  using the same idea
• Notice that the values of m and c will be the same, but different x and y coordinates
• This because it is a different point, on the same line
• For an invariant line under transformation  we can write
• This will create a system of simultaneous equations which can be solved to find the invariant line(s)
• The first equation can be substituted into the second to give an equation in terms of the variable x and the constants m and c
• This equation can then be solved to find the values of m and c by equating the coefficients of x, and then equating the constant terms
• There may be multiple solutions for m and c if there are multiple invariant lines

#### Worked Example

Find the equation of any invariant lines under the transformation