The point P(3, 3) is reflected across the line `y= -x` . Then it is translated horizontally 3 units to the left and vertically 3 units up. Finally, it is reflected across the line `y=x`. What are the coordinates of the point after these transformations ?

Options

A. `(0, -6)`
B. `(0, 0)`
C. `(-6, 6)`
D. `(-6, 0)`

Answer

To solve the problem step by step, let's go through each transformation applied to the point P(3, 3):

### Step 1: Reflect the point P(3, 3) across the line y = -x.

When reflecting a point (α, β) across the line y = -x, the new coordinates become (-β, -α).

For point P(3, 3):
- α = 3
- β = 3

After reflection:
- New coordinates = (-3, -3)

### Step 2: Translate the point horizontally 3 units to the left.

To translate a point (x, y) horizontally to the left by 'a' units, we subtract 'a' from the x-coordinate.

For the point (-3, -3):
- New x-coordinate = -3 - 3 = -6
- y-coordinate remains the same: -3

After translation:
- New coordinates = (-6, -3)

### Step 3: Translate the point vertically 3 units up.

To translate a point (x, y) vertically up by 'b' units, we add 'b' to the y-coordinate.

For the point (-6, -3):
- x-coordinate remains the same: -6
- New y-coordinate = -3 + 3 = 0

After translation:
- New coordinates = (-6, 0)

### Step 4: Reflect the point (-6, 0) across the line y = x.

When reflecting a point (α, β) across the line y = x, the new coordinates become (β, α).

For the point (-6, 0):
- α = -6
- β = 0

After reflection:
- New coordinates = (0, -6)

### Final Result:
The coordinates of the point after all transformations are **(0, -6)**.

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