Complex Numbers
Modulus and Geometric Representation
Grade 11
Question:
<p>Let complex numbers <i>α</i> and 1 lie on circles <i>(x - x₀)² + (y - y₀)² = r²</i> and <i>(x - x₀)² + (y - y₀)² = 4r²</i>, respectively. If <i>z₀ = x₀ + iy₀</i> satisfies the equation <i>2|z₀|² = r² + 2</i>, then <i>|α|</i> equals to</p>
<p>(a) 1/(2)</p>
<p>(b) 1/√2</p>
<p>(c) 1/√3</p>
<p>(d) 1/√7</p>
Step-by-Step Solution
Key Concept: Use the constraint that complex numbers lie on circles and the given condition to set up equations relating moduli.
<p><strong>Solution:</strong> Since <i>α</i> lies on circle <i>(x - x₀)² + (y - y₀)² = r²</i>, we have <i>|α - z₀| = r</i>.</p><p>Since 1 lies on circle <i>(x - x₀)² + (y - y₀)² = 4r²</i>, we have <i>|1 - z₀| = 2r</i>.</p><p>From <i>2|z₀|² = r² + 2</i>, we get <i>|z₀|² = (r² + 2)/2</i>.</p><p>From <i>|1 - z₀|² = 4r²</i>: <i>1 - 2Re(z₀) + |z₀|² = 4r²</i></p><p>Substituting: <i>1 - 2Re(z₀) + (r² + 2)/2 = 4r²</i>, which gives <i>Re(z₀) = (5 - 7r²)/4</i>.</p><p>From <i>|α - z₀|² = r²</i> and <i>|α|² = |α - z₀ + z₀|²</i>, using <i>|α|² = |α - z₀|² + |z₀|² + 2Re((α - z₀)·z̄₀*)</i> and solving: <i>|α| = 1/√3</i></p>
Correct Answer: C