<p>If the differential equation representing the family of all circles touching <em>x</em>-axis at the origin is \((x^2 - y^2)\dfrac{dy}{dx} = g(x)\, y\), then \(g(x)\) equals</p>
Step-by-Step Solution
Key Concept: A circle touching the x-axis at origin has center at (0, r) for some radius r. Differentiate the circle equation x² + (y-r)² = r² to find the differential equation, then eliminate the parameter r using the original circle equation to get g(x).
<p><strong>Step 1:</strong> Set up the circle equation. For a circle touching the x-axis at origin, center is at (0, r). The equation is: x² + (y - r)² = r²</p><p><strong>Step 2:</strong> Expand: x² + y² - 2ry + r² = r², which gives x² + y² = 2ry, so r = (x² + y²)/(2y)</p><p><strong>Step 3:</strong> Differentiate the circle equation x² + (y - r)² = r² implicitly with respect to x:</p><p>2x + 2(y - r)·(dy/dx) = 2r·(dr/dx)</p><p><strong>Step 4:</strong> Since r is a parameter (constant for each circle), dr/dx = 0:</p><p>2x + 2(y - r)·(dy/dx) = 0</p><p>Therefore: (y - r)·(dy/dx) = -x</p><p><strong>Step 5:</strong> Substitute r = (x² + y²)/(2y):</p><p>(y - (x² + y²)/(2y))·(dy/dx) = -x</p><p>((2y² - x² - y²)/(2y))·(dy/dx) = -x</p><p>((y² - x²)/(2y))·(dy/dx) = -x</p><p><strong>Step 6:</strong> Rearrange to match the given form:</p><p>(y² - x²)·(dy/dx) = -2xy</p><p>(x² - y²)·(dy/dx) = 2xy</p><p><strong>Step 7:</strong> Comparing with (x² - y²)·(dy/dx) = g(x)·y, we get: g(x) = 2x</p><p>∴ Answer: C</p>
Correct Answer: C