[4 marks sum]

Question 14 Marks

Points $(3, 0)$ and $(-1, 0)$ are invariant points under reflection in the line $L1$;points $(0, -3)$ and $(0, 1)$ are invariant points on reflection in line $L2$.
(i) Name or write equations for the lines $L1$ and $L2$.
(ii) Write down the images of the points $P (3, 4)$ and $Q (-5, -2)$ on reflection in line $L1$. Name the images as $P’$ and $Q’$ respectively.
(iii) Write down the images of $P$ and $Q$ on reflection in $L2$. Name the images as $P”$ and $Q”$ respectively. (iv) State or describe a single transformation that maps $P’$ onto $p''$

Answer

(i) We know that every point in a line is invariant under the reflection in the same line.Since points $(3,0)$ and $(-1,0)$ lie on the $x$-axis.
So, $(3,0)$ and $(-1,0)$ are invariant under reflection in $x$-axis.
Hence, the equation of line $L_1$ is $y=0$. Similarly, $(0,-3)$ and $(0,1)$ are invariant under reflection in $y$-axis. Hence, the equation of line $L_2$ is $x=0$.
(ii) $P ^{\prime}=$ Image of $P(3,4)$ in $L_1=(3,-4)$
$Q^{\prime}=$ Image of $Q(-5,-2)$ in $L_1=(-5,2)$
(iii) $P ^{\prime \prime}=$ Image of $P (3,4)$ in $L _2=(-3,4)$
$Q^{\prime \prime}=$ Image of $Q(-5,-2)$ in $L_2=(5,-2)$
(iv) Single transformation that maps $P ^{\prime}$ onto $P ^{\prime \prime}$ is reflection in origin.

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Question 24 Marks

P and Q have co-ordinates (0, 5) and (-2, 4).

(a) P is invariant when reflected in an axis. Name the axis.

(b) Find the image of Q on reflection in the axis found in (i).

(d) Write the co-ordinates of the image of Q, obtained by reflecting it in the origin followed by reflection in x-axis.

Answer

(a) Any point that remains unaltered under a given transformation is called an invariant. It is given that P (0, 5) is invariant when reflected in an axis. Clearly, when P is reflected in the y-axis then it will remain invariant. Thus, the required axis is the y-axis.

(b) The co-ordinates of the image of Q (-2, 4) when reflected in y-axis is (2, 4).

(c) (0, k) on reflection in the origin is invariant. We know the reflection of origin in origin is invariant. Thus, k = 0.

(d) Co-ordinates of image of Q (-2, 4) when reflected in origin = (2, -4)

Co-ordinates of image of (2, -4) when reflected in x-axis = (2, 4)

Thus, the co-ordinates of the point are (2, 4).

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Mathematics [4 marks sum]

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