Q.The point $(4,1)$ undergoes the following two successive transformations:

Concept: Coordinate transformations – reflection and translation.

When a point $(a, b)$ is reflected about the line $y = x$, the $x$ and $y$ coordinates swap: $(a, b) \to (b, a)$.

Step 1: Reflect $(4, 1)$ about $y = x$.

The coordinates swap, giving $(1, 4)$.

Step 2: Translate through 2 units along the positive $x$-axis. …

Understanding the transformations

When a point undergoes a reflection about the line $y = x$, the geometric effect is a swap: every point $(a, b)$ maps to $(b, a)$. You can visualize this by imagining the line $y = x$ as a mirror at $45°$ through the origin—points on opposite sides exchange their $x$ and $y$ roles.

Translation along the positive $x$-axis simply means adding to the $x$-coordinate while leaving $y$ unchanged. A translation of 2 units right takes $(p, q)$ to $(p+2, q)$.

We apply these two transformations in sequence to the point $(4, 1)$.

Step-by-step solution

1. Apply the reflection about $y = x$ The original point is $(4, 1)$. Under reflection about $y = x$, we swap the coordinates:

The point is now at $(1, 4)$. …