To solve the problem, we need to analyze the conditions given and understand the geometric implications of the locus of point \( P \) such that \( \angle XPY = 90^\circ \). ### Step-by-Step Explanation 1. **Understanding the Geometry**: - We have two fixed points \( X \) and \( Y \) on a straight line. - The point \( P \) is moving in such a way that the angle formed between the line segments \( XP \) and \( YP \) is always \( 90^\circ \). 2. **Visualizing the Situation**: - Imagine the line segment \( XY \) on a coordinate plane. For simplicity, let’s place \( X \) at the origin (0, 0) and \( Y \) at (a, 0) where \( a \) is a positive distance along the x-axis. - The condition \( \angle XPY = 90^\circ \) means that the point \( P \) must be positioned such that the lines \( XP \) and \( YP \) are perpendicular to each other. 3. **Using the Circle Definition**: - A fundamental property of circles is that if a point \( P \) lies on a circle, the angle subtended by a diameter of that circle at any point on the circle is \( 90^\circ \). - Therefore, if we consider the line segment \( XY \) as the diameter of a circle, any point \( P \) that maintains \( \angle XPY = 90^\circ \) must lie on the circle whose diameter is \( XY \). 4. **Identifying the Locus**: - Since \( P \) can only move on one side of the line \( XY \), the locus of point \( P \) is not the entire circle but rather the upper half of the circle (the semicircle) that is above the line segment \( XY \). 5. **Conclusion**: - Thus, the locus of point \( P \) is a semicircle with \( XY \) as its diameter. ### Evaluating the Options - **Option A: A circle** - This is incorrect because the problem specifies that \( P \) moves on one side of the line \( XY \), which means it cannot cover the entire circle. - **Option B: A semicircle** - This is the correct answer. The locus of point \( P \) is indeed a semicircle above the line segment \( XY \) where \( \angle XPY = 90^\circ \). - **Option C: An arc of a circle through X, Y** - This option is misleading. While the semicircle is indeed an arc of a circle, it does not specify that it is the semicircle above \( XY \). Therefore, it is not as precise as option B. - **Option D: The perpendicular bisector of XY** - This is incorrect because the perpendicular bisector of \( XY \) is a straight line that does not satisfy the condition of maintaining a \( 90^\circ \) angle between \( XP \) and \( YP \) for all points \( P \). ### Summary of Key Points - The locus of point \( P \) such that \( \angle XPY = 90^\circ \) is a semicircle with \( XY \) as the diameter. - The semicircle is located on one side of the line segment \( XY \). - The property of angles subtended by a diameter of a circle is crucial in determining the locus. - Understanding the geometric properties of circles and angles is essential for solving similar problems. This thorough understanding will help you tackle similar questions in your exams effectively!