Circles
Locus Problems
Grade 11
Question:
<p>Let B and C be two fixed points of a given circle and A a variable point lying on major arc of this circle. The locus of feet of perpendiculars dropped from the midpoint of AB on AC is:</p>
<p>(a) Circle such that BC subtends angle \(\tan^{-1}(\tan q)\) in its circumference</p>
<p>(b) Circle such that BC subtends angle \(\tan^{-1}(2 \tan q)\) in its circumference</p>
<p>(c) Circle such that BC subtends angle \(2\tan^{-1}(\tan q)\) in its circumference</p>
<p>(d) Circle such that BC subtends angle \(\cot^{-1}(3 \tan q)\) in its circumference</p>
Step-by-Step Solution
Key Concept: The locus of feet of perpendiculars from the midpoint of a chord to a variable chord forms a circle, with the angle subtended by the fixed chord depending on the geometric configuration.
<p>Using the property that the locus of feet of perpendiculars from a fixed point on a variable line through another fixed point forms a circle, and analyzing the angle subtended by BC, we find the angle to be \(\tan^{-1}(2\tan q)\).</p>
Correct Answer: B