<p>A variable circle passes through the fixed point \(A(p, q)\) and touches \(x\)-axis. The locus of the other end of the diameter through \(A\) is</p>
<p>\((x - p)^2 = 4qy\)</p>
<p>\((x - q)^2 = 4py\)</p>
<p>\((y - p)^2 = 4qx\)</p>
<p>\((y - q)^2 = 4px\)</p>
Step-by-Step Solution
Key Concept: If a circle touches the x-axis, its center lies at (h, r) where r is the radius. Using the condition that A(p,q) lies on the circle and applying the diameter property, we can eliminate the parameter r to find the locus.
<p><strong>Step 1:</strong> Since the circle touches the x-axis, let its center be C(h, r) where r is the radius.</p><p><strong>Step 2:</strong> The circle equation is: (x - h)² + (y - r)² = r²</p><p><strong>Step 3:</strong> Point A(p, q) lies on the circle: (p - h)² + (q - r)² = r²</p><p><strong>Step 4:</strong> Expanding: (p - h)² + q² - 2qr + r² = r², which gives (p - h)² + q² = 2qr</p><p><strong>Step 5:</strong> Therefore: r = [(p - h)² + q²]/(2q)</p><p><strong>Step 6:</strong> If A(p, q) and B(x, y) are endpoints of a diameter through center C(h, r), then: h = (p + x)/2 and r = (q + y)/2</p><p><strong>Step 7:</strong> Substituting into the equation from Step 4: (p - (p+x)/2)² + q² = 2q · (q + y)/2</p><p><strong>Step 8:</strong> Simplifying: ((p - x)/2)² + q² = q(q + y)</p><p><strong>Step 9:</strong> (p - x)²/4 + q² = q² + qy</p><p><strong>Step 10:</strong> (p - x)²/4 = qy</p><p><strong>Step 11:</strong> (x - p)² = 4qy</p><p>∴ Answer: A (The locus is (x - p)² = 4qy, a parabola)</p>
Correct Answer: A