Find the equation of the curve passing through (2,1) which has constant sub-tangent.

Answer

To find the equation of the curve passing through the point (2, 1) with a constant sub-tangent, we will follow these steps:

### Step 1: Understand the concept of sub-tangent
The sub-tangent \( T \) of a curve at a point is given by the formula:
\[ T = \frac{y \, dx}{dy} \]
According to the problem, this sub-tangent is a constant, denoted as \( c \). Thus, we can write:
\[ \frac{y \, dx}{dy} = c \]

### Step 2: Rearrange the equation
Rearranging the equation gives us:
\[ y \, dx = c \, dy \]
Now, we can separate the variables:
\[ \frac{dx}{dy} = \frac{c}{y} \]

### Step 3: Integrate both sides
Integrating both sides with respect to \( y \):
\[ \int dx = \int \frac{c}{y} \, dy \]
This results in:
\[ x = c \ln |y| + k \]
where \( k \) is the constant of integration.

### Step 4: Use the point (2, 1) to find \( k \)
Since the curve passes through the point (2, 1), we can substitute \( x = 2 \) and \( y = 1 \) into the equation:
\[ 2 = c \ln(1) + k \]
Since \( \ln(1) = 0 \), this simplifies to:
\[ 2 = k \]

### Step 5: Substitute \( k \) back into the equation
Now we substitute \( k = 2 \) back into the equation:
\[ x = c \ln |y| + 2 \]

### Step 6: Final equation of the curve
Thus, the equation of the curve is:
\[ x - 2 = c \ln |y| \]
or equivalently,
\[ x = c \ln |y| + 2 \]