If the sum of the distances of point from two perpendicular lines in a plane is 1, then its locus is:

Question

Accepted Answer

To support the given correct answer, Option 3, which states that the locus is a square, let's delve into the reasoning:

1. **Explanation for Option 3 (Square):**
   - Consider a coordinate plane with two perpendicular lines, typically the x-axis and y-axis. The problem states that the sum of the distances from a point (x, y) to these axes is 1. Mathematically, this can be expressed as |x| + |y| = 1.
   - The equation |x| + |y| = 1 represents a geometric figure known as a square. This is because the absolute value function creates linear boundaries that form a square when plotted on a coordinate plane.
   - Specifically, the vertices of this square are at (1,0), (0,1), (-1,0), and (0,-1). The sides of the square are parallel to the coordinate axes, and the square is centered at the origin with its sides aligned along the lines x = ±1 and y = ±1.

2. **Why Option 1 (Circle) is Incorrect:**
   - A circle is defined as the set of all points equidistant from a central point. The equation |x| + |y| = 1 does not describe a circle because the distances involved are not radial from a single center point but rather linear distances from two perpendicular lines. Thus, the locus is not a circle.

3. **Why Option 2 (Straight Line) is Incorrect:**
   - A straight line is described by a linear equation in the form of ax + by = c. The equation |x| + |y| = 1 involves absolute values and represents a closed figure with multiple linear segments, not a single straight line. Therefore, the locus cannot be a straight line.

4. **Why Option 4 (Two Intersecting Lines) is Incorrect:**
   - Two intersecting lines would imply a solution set that consists of only the lines themselves, typically represented by equations like y = mx + c and y = nx + d. The equation |x| + |y| = 1 forms a closed boundary with four segments, not just two intersecting lines. Hence, the locus is not two intersecting lines.

In conclusion, the correct answer is indeed Option 3, as the equation |x| + |y| = 1 forms a square on the coordinate plane, with its sides parallel to the axes and vertices at the specified points.