Combined Transformation

A' be the image obtained when point A(3, -5) is reflected on the line x + y = 0,
A(3, -5) $\ce{->[x \, = \, -y]}$ A'(5, -3)

Again, A" be the final image when point A' is rotated through +90° about the origin,
A'(5, -3) $\ce{->[+90°]}$ A"(3, 5)

Hence, coordinates of the image are A'(5, -3) and A"(3, 5).

Let A(4, 7) be the given point. A' be the image obtained when point A is rotated about the origin through -90°,
A(4, 7) $\ce{->[+90°]}$ A'(7, -4)

Again, A" be the final image when point A' is translated by $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$,
A'(7, -4) $\ce{->[T.V. \begin{pmatrix} 3 \\ 4 \end{pmatrix}]}$ A"(7 + 3, -4 + 4) = A"(10, 0)

Hence, coordinates of the image are A'(7, -4) and A"(10, 0).

P' be the image obtained when point P(2, -3) is reflected on x-axis,
P(2, -3) $\ce{->[x-axis]}$ P'(2, 3)

Again, P" be the final image when point P' is rotated through +90° about origin,
P'(2, 3) $\ce{->[+90°]}$ P"(-3, 2)

Hence, P' is (2, 3) and P" is (-3, 2).

P' be the image obtained when point P(3, 4) is reflected on y-axis,
P(3, 4) $\ce{->[y-axis]}$ P'(-3, 4)

Again, P" be the final image when point P' is rotated through 90° in the positive direction.
P'(-3, 4) $\ce{->[+90°]}$ P"(-4, -3)

Hence, the final image is P"(-4, -3).

P' be the image obtained when point P(-2, 3) is reflected on y-axis,
P(-2, 3) $\ce{->[y-axis]}$ P'(2, 3)

Again, P" be the final image after rotating P' through -90° about the origin.
P'(2, 3) $\ce{->[-90°]}$ P"(3, -2)

Hence, the coordinate of the image formed after the rotation is P"(3, -2)

A' be the image obtained when point A(2, 5) is reflected on x-axis,
A(2, 5) $\ce{->[x-axis]}$ A'(2, -5)

Again, A" be the final image when point A' is rotated through 90° in positive direction about origin.
A'(2, -5) $\ce{->[+90°]}$ A"(5, 2)

Hence, the final image of A is A"(5, 2).

In given combined transformation r₁∘r₂. r₂ is followed by r₁.
r₂ is the reflection on y - 5 = 0, so,
A(x, y) $\ce{->[r_2][\text{y = 5}]}$ A'(x, 10 - y)

r₁ is the reflection on y-axis. so,
A'(x, 10 - y) $\ce{->[r_1][y-axis]}$ A"(-x, 10 - y)

Finally,
A(x, y) $\xrightarrow{r_1 \circ r_2}$ A"(-x, 10 - y)

P" be the final image of point P(5, 3) under the given combination, then,
P(5, 3) $\xrightarrow{r_1 \circ r_2}$ P"(-5, 7)

Let P' be the image obtained when P(5, 6) is reflected on line x = 2,
P(5, 6) $\xrightarrow{\text{x = 2}}$ P'(2 × 2 - 5, 6), P'(-1, 6)

Also, P" be the final image obtained when P' is enlarged by E[(0,0) ,2],
P'(-1, 6) $\xrightarrow{\text{E[(0,0) ,2]}}$ P"(-1 × 2, 6 × 2), P"(-2, 12)

Hence, the coordinate of the images are P'(-1, 6) and P"(-2, 12).

Let A' be the image obtained when A(4, 5) is reflected on line y = 2,
A(4, 5) $\xrightarrow{\text{y = 2}}$ A'(4, 4 - 5), A'(4, -1)

Also, A" be the final image obtained by enlargement of A' on E[(0, 0), 2],
A'(4, -1) $\xrightarrow{\text{E[(0, 0), 2]}}$ A"(8, -2)

Hence, the coordinate of the images are A'(4, -1) and A"(8, -2).

In given combined transformation r₁∘r₂. r₂ is followed by r₁.
r₂ is the rotation about +90° through origin, so,
A(x, y) $\ce{->[r_2][+90°]}$ A'(-y, x)

r₁ is the reflection on x-axis. so,
A'(-y, x) $\ce{->[r_1][x-axis]}$ A"(-y, -x)

Finally,
A(x, y) $\xrightarrow{r_1 \circ r_2}$ A"(-y, -x)

P" be the final image of point P(3, -6) under the given combination, then,
P(3, -6) $\xrightarrow{r_1 \circ r_2}$ P"(6, -3)

We have, R∘T(a , 3) = (5, b) --- (i)
T denotes a translation vector $(\frac23 )$, so,
T(a, 3) $\xrightarrow{\text{T.V.}(\frac 23)}$ T'(a + 2, 6)

From above,
R∘T(a , 3) = R(a + 2, 6)

R denotes the reflection in the line y = 3. so,
R(a + 2, 6) $\xrightarrow{\text{y = 3}}$ R'(a + 2, 0)

So, R∘T(a , 3) = (a + 2, 0)
Hence, by comparing it with (i), we get, a = 3 and b = 0.

Let, A' be the image formed when A(4, -3) is reflected in line x = 0. so,
A(4, -3) $\xrightarrow{\text{x = 0}}$ A'(-4, -3)

Also, A" be the final image formed by reflecting A' in the line y = k. so,
A'(-4, -3) $\xrightarrow{\text{y = k}}$ A"(-4, 2k + 3)

We have given, A"(-4, 9)
Comparing the coordinates of A", we get, 2k + 3 = 9, so, k = 3.

In given combined transformation r₁∘r₂. r₂ is followed by r₁.
r₂ is the rotation about +90° through origin, so,
A(x, y) $\ce{->[r_2][+90°]}$ A'(-y, x)

r₁ is the reflection on x-axis. so,
A'(-y, x) $\ce{->[r_1][x-axis]}$ A"(-y, -x)

Finally,
A(x, y) $\xrightarrow{r_1 \circ r_2}$ A"(-y, -x)

P" be the final image of point P(-3, 2) under the given combination, then,
P(-3, 2) $\xrightarrow{r_1 \circ r_2}$ P"(-2, 3)

In given combined transformation R₁∘R₂. R₂ is followed by R₁.
R₂ is the reflection in the line x = -y, so,
P(x, y) $\ce{->[R_2][\text{x = -y}]}$ P'(-y, -x)

R₁ is the reflection in the line y = 2. so,
P'(-y, -x) $\ce{->[R_1][\text{y = 2}]}$ P"(-y, 4 + x)

Finally,
P(x, y) $\xrightarrow{R_1 \circ R_2}$ P"(-y, -x) --- (I)

Let the point A(a, b) has the image A'(-3, 2) under the given combination, then,
A(a, b) $\xrightarrow{R_1 \circ R_2}$ A'(-3, 2) --- (II)

Comparing the value of a and b by the help of (I) and (II), we get, a = -2 and b = 3

Hence, the required point A(a, b) is A(-2, 3).

Let A(x, y) be the point which is passed in the reflection about the line y = x and A' image is formed.
A(x, y) $\xrightarrow{\text{y = x}}$ A'(y, x)

Again, A" be the image formed when A' is reflected about the line y = 0.
A'(y, x) $\xrightarrow{\text{y = 0}}$ A"(y, -x)

So, from above,
A(x, y) $\rightarrow$ A"(y, -x)

We have given, A"(6, -5). So, comparing A"(y, -x) and A"(6, -5), we get,
x = 5 and y = 6.

Hence, the co-ordinates of point A(x, y) is A(5, 6)