<p>In the diagram, DC is a diameter of the large circle centered at A, and AC is a diameter of the smaller circle centered at B. If DE is tangent to the smaller circle at F and DC = 12, then the length of DE is:</p>
Step-by-Step Solution
Key Concept: Use the property that a tangent to a circle is perpendicular to the radius at the point of tangency, combined with the Pythagorean theorem applied to the right triangle formed by the radius, tangent segment, and chord.
<p><strong>Step 1: Set up coordinates and identify key lengths.</strong></p><p>Given DC = 12 (diameter of large circle centered at A), so radius of large circle = 6.</p><p>Since AC is a diameter of the smaller circle centered at B, and AC is a chord of the large circle, we have AC = 6 (radius of large circle).</p><p>Therefore, radius of small circle = AC/2 = 3, and AB = 3.</p><p><strong>Step 2: Locate key points.</strong></p><p>Place A at origin. Since DC is a diameter of length 12, place D at (-6, 0) and C at (6, 0).</p><p>Since B is the center of the small circle and AC is its diameter with A at origin and C at (6, 0), B is at (3, 0).</p><p>The small circle has center B(3, 0) and radius 3.</p><p><strong>Step 3: Use the tangent condition.</strong></p><p>DE is tangent to the small circle at F. This means BF ⊥ DE, where BF = 3 (radius of small circle).</p><p>Since D is at (-6, 0) and the line DE is tangent to the circle centered at B(3, 0) with radius 3, the distance from B to line DE equals 3.</p><p><strong>Step 4: Find point E on the large circle.</strong></p><p>E lies on the large circle: distance from A to E = 6.</p><p>Let E = (x, y). Then x² + y² = 36.</p><p>Line DE passes through D(-6, 0). The distance from B(3, 0) to line DE is 3.</p><p>If E = (x, y), the line DE has equation: (y - 0)/(x + 6) gives slope m = y/(x + 6).</p><p>Distance from B(3, 0) to line passing through D(-6, 0) with slope m: |m(3 + 6)|/√(1 + m²) = 3.</p><p>This gives |9m|/√(1 + m²) = 3, so 81m² = 9(1 + m²), yielding 72m² = 9, so m² = 1/8, thus m = ±1/(2√2).</p><p><strong>Step 5: Solve for E.</strong></p><p>With slope m = 1/(2√2), line DE: y = [1/(2√2)](x + 6).</p><p>Substituting into x² + y² = 36: x² + [x + 6]²/8 = 36.</p><p>8x² + (x + 6)² = 288, so 8x² + x² + 12x + 36 = 288.</p><p>9x² + 12x - 252 = 0, giving 3x² + 4x - 84 = 0.</p><p>Using the quadratic formula: x = (-4 ± √(16 + 1008))/6 = (-4 ± 32)/6.</p><p>Taking x = 28/6 = 14/3, then y = [1/(2√2)](14/3 + 6) = [1/(2√2)](32/3) = 16/(3√2) = 8√2/3.</p><p><strong>Step 6: Calculate DE.</strong></p><p>DE = √[(14/3 + 6)² + (8√2/3)²] = √[(32/3)² + 128/9] = √[1024/9 + 128/9] = √(1152/9) = √128 = 8√2.</p><p><strong>∴ Answer: A</strong></p>
Correct Answer: A