Straight Lines
Locus
Grade 11
Question:
<p>Let A = (0,0), B = (4,0) and on segment AB is given a point M. On the same side of AB, squares AMCD and BMFE are constructed above AB. The circumcircles S₁ and S₂ of two squares AMCD and BMFE respectively have centres P and Q.</p><p>The locus of midpoints of all segments PQ as M varies along the segment AB is:</p>
<p>(a) line segment (x,1), x ∈ [1,3]</p>
<p>(b) line segment (x,1), x ∈ [2,4]</p>
<p>(c) line segment (x,1), x ∈ [0,3]</p>
<p>(d) line segment (x,1), x ∈ [0,4]</p>
Step-by-Step Solution
Key Concept: Find the positions of circumcenters P and Q, then track their midpoint as M moves along AB.
<p><strong>Analysis:</strong> The center P of square AMCD is at (AM/2, AM/2), and the center Q of square BMFE is at ((AM + (4-AM))/2, (4-AM)/2). The midpoint of PQ traces a horizontal line segment. When M varies from A to B, the midpoint locus is the line segment (x,1) where x ∈ [0,4].</p>
Correct Answer: d