Finding the Locus of the Midpoint

The question asks for the equation of the locus of the mid-point of a line segment. This segment is formed when the line given by the equation \(x + y = p\) intersects the coordinate axes. Here, \(p\) is a real number, which means the position of the line changes depending on the value of \(p\).

Step 1: Find the points of intersection with the coordinate axes

The coordinate axes are the x-axis and the y-axis. A point on the x-axis has a y-coordinate of 0, and a point on the y-axis has an x-coordinate of 0.

• Intersection with the x-axis: Set \(y = 0\) in the equation \(x + y = p\).
  \(x + 0 = p\)
  \(x = p\)
  The point of intersection with the x-axis is \((p, 0)\). Let's call this point A.
• Intersection with the y-axis: Set \(x = 0\) in the equation \(x + y = p\).
  \(0 + y = p\)
  \(y = p\)
  The point of intersection with the y-axis is \((0, p)\). Let's call this point B.

The line segment is the part of the line between points A\((p, 0)\) and B\((0, p)\).

Step 2: Find the coordinates of the midpoint of the segment AB

Let the mid-point of the line segment AB be denoted by \(M(h, k)\). The formula for the midpoint of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \).

Using points A\((p, 0)\) and B\((0, p)\):

• The x-coordinate of the midpoint is \(h = \frac{p + 0}{2} = \frac{p}{2}\).
• The y-coordinate of the midpoint is \(k = \frac{0 + p}{2} = \frac{p}{2}\).

So, the coordinates of the midpoint \(M\) are \(\left(\frac{p}{2}, \frac{p}{2}\right)\).

Step 3: Find the locus of the midpoint

The locus of the midpoint is the path traced by the point \((h, k)\) as the parameter \(p\) varies. We have the equations:

\(h = \frac{p}{2}\)     (Equation 1)

\(k = \frac{p}{2}\)      (Equation 2)

To find the relationship between \(h\) and \(k\) that defines the locus, we need to eliminate the parameter \(p\). From Equation 1, we can express \(p\) in terms of \(h\):

\(p = 2h\)

Substitute this expression for \(p\) into Equation 2:

\(k = \frac{2h}{2}\)

\(k = h\)

This equation \(k = h\) gives the relationship between the coordinates of the midpoint \((h, k)\). To find the equation of the locus, we replace \(h\) with \(x\) and \(k\) with \(y\).

So, the equation of the locus of the midpoint is \(y = x\), which can be written as \(x - y = 0\).

This equation represents a straight line passing through the origin with a slope of 1.

Conclusion

The equation of the locus of the mid-point of the line segment obtained by cutting the line \(x + y = p\) by the coordinate axes is \(x - y = 0\).

Revision Table: Key Concepts

Additional Information: Understanding Locus

Finding the locus of a point involves identifying a moving point that satisfies certain geometrical conditions and then determining the equation that describes the path traced by this point. The process often involves:

• Assigning coordinates, often generic variables like \((h, k)\) or \((x, y)\), to the moving point whose locus is required.
• Writing down the given conditions in terms of these coordinates and any other variables (like \(p\) in this case).
• Eliminating any parameters (variables that change but are not part of the final equation) to get an equation solely in terms of the coordinates of the moving point.
• Replacing the generic variables (like \(h, k\)) with standard coordinate variables \((x, y)\) to get the final equation of the locus.

In this problem, the condition is that the point \((h, k)\) is the midpoint of the segment cut by the axes from the line \(x+y=p\). By expressing \((h, k)\) in terms of \(p\) and then eliminating \(p\), we found the relationship between \(h\) and \(k\), which is the equation of the locus.