The perpendicular from the origin to the tangent at any point on a curve is equal to the abscissa of the point of contact. Also curve passes through the point $(1, 1)$. Then the length of intercept of the curve on the x-axis is ______.
Step-by-Step Solution
Key Concept: The perpendicular distance from the origin to a tangent line determines the geometric constraint that yields a differential equation.
The equation of the tangent line at $(x,y)$ is $X\frac{dy}{dx} - Y\frac{dy}{dx} + y = 0$. The perpendicular distance from the origin to this tangent line is given by the formula for distance from a point to a line. Setting this equal to a constant or given condition would yield the required differential equation relating $y$, $\frac{dy}{dx}$, and $x$.
Correct Answer: 2