CAIE A-Level Further Mathematics 1.4.5 Invariant Points and Lines

Practise transforming a general point, imposing the invariant-line or invariant-point condition and solving for complete line equations rather than gradients alone.

Exam points

write the transformed coordinates, set y = mx and impose Y = mX for a line through the origin
solve the resulting equation for m and state each full invariant-line equation, not only its gradient
for invariant points, equate transformed and original coordinates and solve the simultaneous system

[Maximum number: 5]

Let k be a constant. The matrices A, B and C are given by

\mathbf{A}=\left(\begin{array}{lll} 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rr} 0 & -2 \\ -1 & 3 \\ 0 & 0 \end{array}\right) \quad \text { and } \quad \mathbf{C}=\left(\begin{array}{rrr} -2 & -1 & 1 \\ 1 & 1 & 3 \end{array}\right)

It is given that A is singular.

Find the equations of the invariant lines, through the origin, of the transformation in the x-y plane represented by CAB.

$\left(\begin{array}{cc}3 -7 -9 3\end{array}\right)\binom{x}{y}=\binom{3 x-7 y}{-9 x+3 y}$

B1

Transforms $\binom{x}{y}$ to $\binom{X}{Y}$ .

-9 x+3 m x=m(3 x-7 m x)

M1 A1

Uses y=m x and Y=m X.

$-9+3 m=3 m-7 m^{2} \Rightarrow 7 m^{2}=9$

A1

$y=\frac{3}{\sqrt{7}} x$ and $y=-\frac{3}{\sqrt{7}} x$

A1

5

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1.4.5—Invariant points and lines question 1