Q.Let $R$ be set of points inside a rectangle of sides $a$ and $b$ $(a, b > 1)$ with two sides along the positive direction of $x$-axis and $y$-axis. Then
(A) $R = \{(x, y) : 0 \le x \le a,\ 0 \le y \le b\}$
(B) $R = \{(x, y) : 0 \le x < a,\ 0 \le y \le b\}$
(C) $R = \{(x, y) : 0 \le x \le a,\ 0 < y < b\}$
(D) $R = \{(x, y) : 0 < x < a,\ 0 < y < b\}$

The key idea is the precise definition of "inside" a geometric region in coordinate geometry.

A rectangle with two sides along the positive $x$-axis and $y$-axis, and side lengths $a$ and $b$, has its vertices at $(0,0)$, $(a,0)$, $(0,b)$, and $(a,b)$.

The set of points inside this rectangle refers to its interior, which strictly excludes all boundary points (the sides and vertices). …

Concept and Intuition

When we talk about the "inside" of a geometric shape, we are referring to the interior of that shape. This is a crucial distinction from the shape itself, which often includes its boundary. For example, a circle includes its circumference, but the "inside of a circle" refers only to the points strictly within the circumference, not on it.

In coordinate geometry, this distinction is represented by the type of inequality used:

• Strict inequalities ($<$ or $>$): These define points that are not on the boundary. For instance, $x < a$ means $x$ can be any value up to $a$, but not $a$ itself.
• Non-strict inequalities ($\le$ or $\ge$): These define points that include the boundary. For instance, $x \le a$ means $x$ can be any value up to $a$, including $a$ itself.

For a rectangle, its boundary consists of four line segments. If a point is "inside" the rectangle, it cannot lie on any of these four segments. It must be strictly between the minimum and maximum x-coordinates, and strictly between the minimum and maximum y-coordinates.

Step-by-Step Solution

1. Understand the Rectangle's Position:

   The problem states that the rectangle has two sides along the positive direction of the $x$-axis and $y$-axis. This means one vertex of the rectangle is at the origin $(0,0)$. Since the sides have lengths $a$ and $b$, the vertices of the rectangle are $(0,0)$, $(a,0)$, $(0,b)$, and $(a,b)$.

2. Define the Range for x-coordinates:

   For any point $(x,y)$ to be inside this rectangle, its $x$-coordinate must be greater than the $x$-coordinate of the left boundary (which is $0$) and less than the $x$-coordinate of the right boundary (which is $a$). If $x$ were $0$ or $a$, the point would be on one of the vertical boundary lines, not strictly inside.

   Therefore, for points inside the rectangle, we must have $0 < x < a$.

3. Define the Range for y-coordinates:

   Similarly, for any point $(x,y)$ to be inside this rectangle, its $y$-coordinate must be greater than the $y$-coordinate of the bottom boundary (which is $0$) and less than the $y$-coordinate of the top boundary (which is $b$). If $y$ were $0$ or $b$, the point would be on one of the horizontal boundary lines, not strictly inside.

   Therefore, for points inside the rectangle, we must have $0 < y < b$.

4. Combine the Conditions:

   To be inside the rectangle, a point $(x,y)$ must satisfy both conditions simultaneously.

   So, the set $R$ of points inside the rectangle is given by:

   $R = \{(x, y) : 0 < x < a,\ 0 < y < b\}$

5. Compare with Given Options:

   Let's examine the given options: …