Circles
Tangent Circles
Grade 11
Question:
<p>Given a line segment AB, where A = (0,0) and B = (a,0). Three circles S₁, S₂, S₃ of radius R are centred at the end points and the midpoint of the line segment AB. A fourth circle S₄ is drawn touching the 3 given circles.</p><p>If \(\frac{a}{4} < R < \frac{a}{2}\), then radius of circle S₄ is:</p>
<p>(a) \(\frac{a^2}{16R}\)</p>
<p>(b) \(\frac{a^2}{8R}\)</p>
<p>(c) \(\frac{3a^2}{16R}\)</p>
<p>(d) \(\frac{a^2}{4R}\)</p>
Step-by-Step Solution
Key Concept: Recognize that the formula for the radius changes as the parameter R moves into a different range.
<p><strong>Analysis:</strong> When \(\frac{a}{4} < R < \frac{a}{2}\), the geometric configuration changes. The radius of the circle S₄ tangent to all three circles is given by \(\frac{3a^2}{16R}\).</p>
Correct Answer: c