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: The locus is found by parametrizing the midpoint coordinates in terms of the parameter m and finding the range of x-coordinates.
<p><strong>Step 1:</strong> Let M = (m, 0) where m ∈ [0, 4].</p><p><strong>Step 2:</strong> The centre P of square AMCD is at P = (m/2, m/2).</p><p><strong>Step 3:</strong> The centre Q of square BMFE is at Q = ((4+m)/2, (4-m)/2).</p><p><strong>Step 4:</strong> The midpoint of PQ is: \(\left(\frac{\frac{m}{2} + \frac{4+m}{2}}{2}, \frac{\frac{m}{2} + \frac{4-m}{2}}{2}\right) = \left(\frac{4+2m}{4}, \frac{4}{4}\right) = \left(\frac{2+m}{2}, 1\right) = \left(1 + \frac{m}{2}, 1\right)\)</p><p><strong>Step 5:</strong> As m varies from 0 to 4, \(1 + \frac{m}{2}\) varies from 1 to 3.</p><p><strong>Step 6:</strong> Therefore, the locus is the line segment (x, 1) where x ∈ [1, 3].</p><p>∴ Answer is (a).</p>
Correct Answer: A