Plz answer the question..16A and B are two fixed points;if PA and PB intersect a constant distance 2c from a given straight line , find the locus of P.

To find the locus of point P, given that the distances from points A and B to point P are always equal to a constant distance of 2c from a given straight line, we can approach this problem step by step. Let's break it down clearly.

Understanding the Setup

We have two fixed points, A and B, and a straight line. The point P can move around, but it must maintain a specific relationship with the line and the points A and B. The condition states that the distances from P to the line are always 2c.

Visualizing the Problem

Imagine the straight line as a horizontal line on a coordinate plane. The points A and B can be positioned anywhere in relation to this line. The distance from point P to the line is measured perpendicularly. Therefore, if we denote the line as y = k (where k is a constant), the distance from point P (x, y) to the line is given by the formula:

• Distance from P to the line = |y - k|

According to the problem, this distance must equal 2c, which gives us the equation:

• |y - k| = 2c

Finding the Locus of P

This absolute value equation implies two scenarios:

• y - k = 2c → y = k + 2c
• y - k = -2c → y = k - 2c

Thus, the locus of point P, in terms of y, consists of two horizontal lines: one at y = k + 2c and another at y = k - 2c. These lines are parallel to the original line and are located at a distance of 2c above and below it.

Considering Points A and B

Now, let's incorporate the fixed points A and B. The condition that PA and PB intersect at a constant distance from the line suggests that the locus of P is not just limited to the two horizontal lines we derived. Instead, we need to consider the geometric relationship between points A, B, and the lines.

The locus of P will actually form a shape that is symmetric with respect to the line segment AB. This means that as P moves along the lines y = k + 2c and y = k - 2c, it will also maintain a certain distance from the segment connecting A and B.

Final Locus Description

In conclusion, the locus of point P is defined by two horizontal lines parallel to the given straight line, located at y = k + 2c and y = k - 2c. However, if we consider the distances PA and PB, the actual path traced by P will be constrained by the positions of A and B, leading to a more complex geometric figure, potentially an ellipse or hyperbola, depending on the relative positions of A and B.

To summarize, the locus of P consists of two horizontal lines, but the final shape may vary based on the specific locations of points A and B. This relationship highlights the interplay between fixed points and the dynamic nature of point P as it moves while adhering to the given constraints.