To find the differential equation of the curve for which the point of tangency divides the segment of the tangent between the coordinate axes in the ratio \(1:2\), we can follow these steps:
### Step 1: Define the Point of Tangency
Let the point of tangency on the curve be \(P(x_1, y_1)\).
### Step 2: Write the Equation of the Tangent Line
The equation of the tangent line at point \(P\) can be expressed as:
\[
y - y_1 = m(x - x_1)
\]
where \(m\) is the slope of the tangent line, which is given by \(\frac{dy}{dx}\) at the point \(P\).
### Step 3: Find the Intercepts of the Tangent Line
To find the x-intercept (let's call it \(A\)), set \(y = 0\):
\[
0 - y_1 = m(x - x_1) \implies x = x_1 - \frac{y_1}{m}
\]
Thus, the coordinates of point \(A\) are:
\[
A\left(x_1 - \frac{y_1}{m}, 0\right)
\]
For the y-intercept (let's call it \(B\)), set \(x = 0\):
\[
y - y_1 = m(0 - x_1) \implies y = y_1 - mx_1
\]
Thus, the coordinates of point \(B\) are:
\[
B(0, y_1 - mx_1)
\]
### Step 4: Use the Section Formula
The point \(P(x_1, y_1)\) divides the segment \(AB\) in the ratio \(1:2\). According to the section formula, the coordinates of point \(P\) can be expressed as:
\[
P\left(\frac{2 \cdot 0 + 1 \left(x_1 - \frac{y_1}{m}\right)}{1 + 2}, \frac{2\left(y_1 - mx_1\right) + 1 \cdot 0}{1 + 2}\right)
\]
This simplifies to:
\[
P\left(\frac{x_1 - \frac{y_1}{m}}{3}, \frac{2\left(y_1 - mx_1\right)}{3}\right)
\]
### Step 5: Set Up the Equation
From the section formula, we can equate the coordinates:
1. For the x-coordinate:
\[
x_1 = \frac{x_1 - \frac{y_1}{m}}{3} \implies 3x_1 = x_1 - \frac{y_1}{m} \implies 2x_1 = -\frac{y_1}{m} \implies m = -\frac{y_1}{2x_1}
\]
2. For the y-coordinate:
\[
y_1 = \frac{2\left(y_1 - mx_1\right)}{3} \implies 3y_1 = 2y_1 - 2mx_1 \implies y_1 = -2mx_1
\]
### Step 6: Substitute for \(m\)
Substituting \(m = -\frac{y_1}{2x_1}\) into \(y_1 = -2mx_1\):
\[
y_1 = -2\left(-\frac{y_1}{2x_1}\right)x_1 \implies y_1 = y_1
\]
This confirms the relationship holds.
### Step 7: Form the Differential Equation
Now, substituting \(m = \frac{dy}{dx}\):
\[
\frac{dy}{dx} = -\frac{y}{2x}
\]
Cross-multiplying gives:
\[
2y \, dx + x \, dy = 0
\]
### Final Result
Thus, the differential equation of the curve is:
\[
x \, dy + 2y \, dx = 0
\]
