Question:

Step 1: Let the point of tangency on the curve be $(x_0, y_0)$. The equation of the tangent line to the curve at this point is given by: $$Y - y_0 = \frac{dy}{dx}\Big|_{(x_0, y_0)} (X - x_0)$$ Let $m = \frac{dy}{dx}\Big|_{(x_0, y_0)}$. The equation becomes: $$Y - y_0 = m(X - x_0)$$ To find the x-intercept, set $Y=0$: $$-y_0 = m(X - x_0) \implies X = x_0 - \frac{y_0}{m}$$ The x-intercept is $\left(x_0 - \frac{y_0}{m}, 0\right)$. To find the y-intercept, set $X=0$: $$Y - y_0 = m(-x_0) \implies Y = y_0 - mx_0$$ The y-intercept is $\left(0, y_0 - mx_0\right)$. Step 2: The problem states that the segment of the tangent lying between the coordinate axes is bisected by the point of tangency $(x_0, y_0)$. The midpoint of the segment connecting the x-intercept and y-intercept is: $$\left(\frac{\left(x_0 - \frac{y_0}{m}\right) + 0}{2}, \frac{0 + (y_0 - mx_0)}{2}\right)$$ This midpoint must be equal to $(x_0, y_0)$. Equating the coordinates: $$\frac{x_0 - \frac{y_0}{m}}{2} = x_0 \quad \text{and} \quad \frac{y_0 - mx_0}{2} = y_0$$ From the x-coordinate: $$x_0 - \frac{y_0}{m} = 2x_0 \implies -\frac{y_0}{m} = x_0 \implies m = -\frac{y_0}{x_0}$$ From the y-coordinate: $$y_0 - mx_0 = 2y_0 \implies -mx_0 = y_0 \implies m = -\frac{y_0}{x_0}$$ Both conditions yield the same differential equation: $$\frac{dy}{dx} = -\frac{y}{x}$$ Step 3: Solve the differential equation: $$\frac{dy}{y} = -\frac{dx}{x}$$ Integrate both sides: $$\int \frac{dy}{y} = -\int \frac{dx}{x}$$ $$\ln|y| = -\ln|x| + C_1$$ $$\ln|y| + \ln|x| = C_1$$ $$\ln|xy| = C_1$$ $$|xy| = e^{C_1}$$ Let $k = \pm e^{C_1}$, where $k$ is a non-zero constant. $$xy = k$$ The curve passes through the point $(1, 3)$. Substitute these values into the equation: $$1 \cdot 3 = k \implies k = 3$$ Therefore, the equation of the curve is: $$xy = 3$$