Geometric Applications

Question

The length of the subtangent at any point of a curve is constant and equal to a, leading to a differential equation whose solution is the curve family sought here.

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The subtangent at a point on a curve is the projection on the x-axis of the segment of the tangent between the point and the x-axis, given by the formula \frac{y}{dy/dx}. The conceptual point is that geometric tangent quantities translate into differential equations through this standard subtangent expression. Setting the subtangent equal to the constant a gives \frac{y}{dy/dx} = a, which rearranges to \frac{dy}{dx} = \frac{y}{a}. This is a separable, exponential-growth-type equation: \frac{dy}{y} = \frac{dx}{a}. Integrating both sides yields \ln|y| = \frac{x}{a} + C_1, and exponentiating gives y = Ce^{x/a}. Option y = Ce^{-x/a} would arise from a subtangent of -a, contradicting the positive constant. Option y = a\ln x + C corresponds to a different geometric condition involving the subnormal or area. Option y = Cx^a has slope that does not produce a constant subtangent. This is the standard JEE Advanced subtangent application. As a final check, for y = Ce^{x/a} we have \frac{dy}{dx} = \frac{C}{a}e^{x/a} = \frac{y}{a}, so the subtangent \frac{y}{y/a} = a is indeed constant everywhere, confirming the result.

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