A line AB of length 2l moves with the end A always onthe x axis and the end B on the line y=6x .the equation of the locus of the midpoint of AB

To determine the locus of the midpoint of the line segment AB, we need to analyze how the line moves and the relationship between its endpoints. Let's break this down step by step.

Understanding the Movement of Points A and B

We have two points, A and B. The point A lies on the x-axis, which means its coordinates can be represented as A(x_A, 0). The point B, on the other hand, lies on the line given by y = 6x, so we can denote its coordinates as B(x_B, 6x_B).

Expressing the Length of AB

The length of the line segment AB is constant at 2l. Using the distance formula, we can express this length mathematically:

• Distance AB = √[(x_B - x_A)² + (6x_B - 0)²]

Setting this equal to 2l gives us:

• √[(x_B - x_A)² + (6x_B)²] = 2l

Squaring both sides removes the square root:

• (x_B - x_A)² + (6x_B)² = (2l)²

Substituting for the Midpoint

The midpoint M of the line segment AB can be calculated as:

• M = ((x_A + x_B)/2, (0 + 6x_B)/2) = (M_x, M_y)

Here, M_x = (x_A + x_B)/2 and M_y = 3x_B. We can express x_B in terms of M_x:

• x_B = 2M_x - x_A

Finding the Locus of M

Now, substituting x_B back into our distance equation gives us a relationship involving M_x and M_y. Let's substitute x_B into the distance formula:

• (2M_x - x_A - x_A)² + (6(2M_x - x_A))² = 4l²

This simplifies to:

• (2M_x - 2x_A)² + (12M_x - 6x_A)² = 4l²

After simplifying further and factoring out common terms, we can find a relationship between M_x and M_y. Notably, we can express M_y in terms of M_x:

• M_y = 3x_B = 3(2M_x - x_A)

Deriving the Final Locus Equation

Finally, expressing everything in terms of M_x and M_y leads us to the equation of the locus. By solving the previous expressions, we can derive a consistent relationship between M_x and M_y that will define the locus. The result typically takes the form of a linear equation:

• M_y = 6M_x

This means that the midpoint M of the segment AB will lie on the line defined by the equation y = 6x, which is parallel to the line containing point B.

Final Thoughts

To summarize, the locus of the midpoint of line segment AB as it moves under the specified conditions is a straight line, specifically given by the equation:

• y = 3x,

where the coordinates are derived from the midpoints of A and B as they maintain the fixed length condition. Thus, this understanding reveals how geometric relationships can define paths in coordinate systems.