<p>The centres of those circles which touch the circle, \(x^2 + y^2 - 8x - 8y - 4 = 0\), externally and also touch the \(x\)-axis, lie on</p>
Step-by-Step Solution
Key Concept: If a circle touches the x-axis, its center is at (h, r) where r is the radius. Use the external tangency condition: distance between centers equals sum of radii to set up a constraint equation that the locus must satisfy.
<p><strong>Step 1:</strong> Rewrite the given circle in standard form.</p><p>x² + y² - 8x - 8y - 4 = 0</p><p>(x - 4)² + (y - 4)² = 36</p><p>Center C₁ = (4, 4), Radius R = 6</p><p><strong>Step 2:</strong> Let the required circle have center P(h, k) and radius r. Since it touches the x-axis: k = r (center is at distance r from x-axis)</p><p><strong>Step 3:</strong> For external tangency with the given circle:</p><p>|PC₁| = R + r</p><p>√[(h - 4)² + (k - 4)²] = 6 + r</p><p><strong>Step 4:</strong> Substitute k = r:</p><p>√[(h - 4)² + (r - 4)²] = 6 + r</p><p><strong>Step 5:</strong> Square both sides:</p><p>(h - 4)² + (r - 4)² = (6 + r)²</p><p>(h - 4)² + r² - 8r + 16 = 36 + 12r + r²</p><p>(h - 4)² = 36 + 12r + 8r - 16</p><p>(h - 4)² = 20 + 20r</p><p>(h - 4)² = 20(1 + r)</p><p><strong>Step 6:</strong> Replace h with x and r with y (since center is (h, k) = (x, y)):</p><p>(x - 4)² = 20(1 + y)</p><p>(x - 4)² = 20y + 20</p><p>∴ Answer: A (The locus is the parabola (x - 4)² = 20(y + 1) or equivalent form)</p>
Correct Answer: A