Circles
Apollonius circle and locus
Grade 11
Question:
<p><span class="math inline">\(A\)</span>, <span class="math inline">\(B\)</span> and <span class="math inline">\(C\)</span> are points in the xy-plane such that <span class="math inline">\(A(1, 2)\)</span>; <span class="math inline">\(B(5, 6)\)</span> and <span class="math inline">\(AC = 3BC\)</span>. Then:</p>
<p>(a) ABC is a unique triangle</p>
<p>(b) There can be only two such triangles</p>
<p>(c) No such triangle is possible</p>
<p>(d) There can be infinite number of such triangles</p>
Step-by-Step Solution
Key Concept: The locus of points with a constant ratio of distances to two fixed points is a circle (Apollonius circle), which contains infinitely many points.
<p><strong>Solution:</strong> The condition <span class="math inline">\(AC = 3BC\)</span> means point <span class="math inline">\(C\)</span> lies on a circle (or circles) determined by the ratio of distances from fixed points <span class="math inline">\(A\)</span> and <span class="math inline">\(B\)</span>. This is the Apollonius circle. Since <span class="math inline">\(C\)</span> can be any point on this circle (or locus), there are infinitely many possible positions for <span class="math inline">\(C\)</span>, hence infinitely many triangles.</p>
Correct Answer: D