Invariant Points & Lines (Edexcel A-Level Further Mathematics): Revision Notes

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2.2.3 Invariant Points & Lines

Overview

Invariant Points

A point is invariant under a linear transformation if its coordinates remain unchanged after the transformation is applied. Mathematically, for a matrix MM, a point x=(x,y)\mathbf{x} = (x, y) is invariant if Mx=xM \mathbf{x} = \mathbf{x}

Invariant Lines

A line is invariant under a linear transformation if any point on the line is transformed to another point on the same line. For a line y=mx+cy = mx + c, it is invariant if:

M(xy)=(x′y′),M \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix},

where y′=mx′+cy′=mx′+c

Finding Invariant Points

Start with the Given Transformation Matrix MM:

Write the transformation equations:

(x′y′)=M(xy).\begin{pmatrix} x' \\ y' \end{pmatrix} = M \begin{pmatrix} x \\ y \end{pmatrix}.

Equate this to (xy)\begin{pmatrix} x \\ y \end{pmatrix}

M(xy)=(xy).M \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}.

This gives two equations to solve.

Simplify and Solve the Equations:

Rearrange the equations to isolate xx and yy, then solve simultaneously.

Interpret the Results:

  • A single solution corresponds to a specific invariant point.
  • Infinite solutions describe a line of invariant points.

Finding Invariant Lines

Start with the General Line Equation:

Use:

y=mx+cy = mx + c

Substitute into the Transformation:

Replace yy with mx+cmx+c in the transformation matrix:

(x′y′)=M(xmx+c)\begin{pmatrix} x' \\ y' \end{pmatrix} = M \begin{pmatrix} x \\ mx + c \end{pmatrix}

Simplify and Equate:

The transformed y′y′ must satisfy y′=mx′+cy′=mx′+c. Rearrange to form equations for mm and cc.

Analyze the Conditions:

Solve for mm (the slope) and cc (the intercept) based on consistent equations.

Worked Examples

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Example 1: Finding Invariant Points

Given Matrix

Let:

M=(4.21.61.61.8)M = \begin{pmatrix} 4.2 & 1.6 \\ 1.6 & 1.8 \end{pmatrix}

Step 1: Write the Transformation Equations

From

M(xy)=(xy)M \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}

We get:

4.2x+1.6y=x,1.6x+1.8y=y.\begin{aligned} 4.2x + 1.6y &= x, \\ 1.6x + 1.8y &= y. \end{aligned}

Step 2: Rearrange Equations

Rewriting each equation:

3.2x+1.6y=0(Equation 1),3.2x + 1.6y = 0 \quad \text{(Equation 1)},1.6x+0.8y=0(Equation 2).1.6x + 0.8y = 0 \quad \text{(Equation 2)}.

Step 3: Solve Simultaneously

Divide Equation 22 by 0.80.8:

2x+y=0⇒y=−2x.2x + y = 0 \quad \Rightarrow \quad y = -2x.

Step 4: Interpret the Solution

The invariant points form the line:

y=−2x.y = -2x.
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Example 2: Finding Invariant Lines

Given Matrix

Let:

M=(72424−7).M = \begin{pmatrix} 7 & 24 \\ 24 & -7 \end{pmatrix}.

Step 1: Start with General Line Equation

Assume:

y=mx+c.y = mx + c.

Step 2: Apply the Transformation

Substitute y=mx+cy = mx + c into the transformation:

(x′y′)=M(xmx+c).\begin{pmatrix} x' \\ y' \end{pmatrix} = M \begin{pmatrix} x \\ mx + c \end{pmatrix}.

This expands to:

x′=7x+24(mx+c),y′=24x−7(mx+c).\begin{aligned} x' &= 7x + 24(mx + c), \\ y' &= 24x - 7(mx + c). \end{aligned}

Step 3: Expand and Equate

Using y′=mx′+cy' = mx' + c, substitute x′x' and y′y':

24x−7(mx+c)=m(7x+24mx+24c)+c.24x - 7(mx + c) = m(7x + 24mx + 24c) + c.

Step 4: Simplify

Gather terms involving xx and cc:

0=2x(12m+7m−12)+8c(1+3m).0 = 2x(12m + 7m - 12) + 8c(1 + 3m).

Step 5: Solve for mm and cc

If c=0c = 0, the quadratic simplifies, giving:

m=43orm=−34.m = \frac{4}{3} \quad \text{or} \quad m = -\frac{3}{4}.

Step 6: Write the Invariant Lines

The invariant lines are:

y=43xandy=−34x.y = \frac{4}{3}x \quad \text{and} \quad y = -\frac{3}{4}x.

Note Summary

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Common Mistakes

  1. Confusing Invariant Points with Invariant Lines: Invariant points remain unchanged, while invariant lines may map to themselves but change points along the line.

  2. Incorrect Setup: Forgetting to use Mx=xM \mathbf{x} = \mathbf{x} for points or y′=mx′+cy′=mx′+c for lines.

  3. Special Cases: Missing c = 0 or symmetrical properties of the transformation matrix.

  4. Verification Errors: Failing to substitute solutions back into the original equations.

  5. Quadratic Errors: Miscalculating the values of mm or cc.

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Key Formulas

  1. Invariant Point Condition:
M(xy)=(xy).M \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ y \end{pmatrix}.

  1. Invariant Line General Form: y=mx+cy=mx+c, where y′=mx′+cy′=mx′+c

  2. Matrix Transformation for Line Verification:

M(xmx+c)=(x′y′)M \begin{pmatrix} x \\ mx + c \end{pmatrix} = \begin{pmatrix} x' \\ y' \end{pmatrix}
  1. Slope Conditions: Solve the quadratic equation derived from MM:
0=2x(12m+7m−12)+8c(1+3m).0 = 2x(12m + 7m - 12) + 8c(1 + 3m).
  1. Intercept Conditions: If c = 0, this simplifies to specific invariant lines:
y=43x or y=−34x.y = \frac{4}{3}x\ \text{or}\ y = -\frac{3}{4}x.

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