Circles
Chord and Midpoint Locus
Grade 11
Question:
<p>Let C be a circle of radius r with centre at O, let P be a point outside C and D be a point on C. A line through P intersects C at Q and R, S is the midpoint of QR. For different choices of lines through P, what is the curve on which S lies:</p>
<p>(a) a straight line</p>
<p>(b) an arc of circle with P as centre</p>
<p>(c) an arc of circle with PS as diameter</p>
<p>(d) an arc of circle with OP diameter</p>
Step-by-Step Solution
Key Concept: The midpoint of a chord is collinear with the centre and perpendicular to the chord. The locus of points forming a right angle with a fixed diameter traces a circle.
<p><strong>Analysis:</strong> Since S is the midpoint of chord QR, and O is the centre of circle C, the line OS is perpendicular to QR. As the line through P varies, S traces a curve such that ∠OSP = 90°. By the locus property, if OS ⊥ PS for all positions, then S lies on a circle with diameter OP.</p>
Correct Answer: d