The equation of the curve which passes through the point (2a, a) and for which the sum of the Cartesian sub tangent and the abscissa is equal to the constant a, is:

Options

A. `y(x-a)=a^2`
B. `y(x+a)=a^2`
C. `x(y-a)=a^2`
D. `x(y+a)=a^2`

Answer

To solve the problem, we need to find the equation of the curve that passes through the point (2a, a) and satisfies the condition that the sum of the Cartesian sub-tangent and the abscissa is equal to the constant \( a \).

### Step 1: Understand the condition
The condition states that the sum of the Cartesian sub-tangent and the abscissa is equal to \( a \). The Cartesian sub-tangent \( T \) is given by:
\[ T = \frac{y}{\frac{dy}{dx}} \]
Thus, the condition can be expressed as:
\[ T + x = a \]
Substituting for \( T \):
\[ \frac{y}{\frac{dy}{dx}} + x = a \]
Rearranging gives:
\[ \frac{y}{\frac{dy}{dx}} = a - x \]

### Step 2: Rearranging the equation
We can rearrange the equation to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{y}{a - x} \]

### Step 3: Separate variables
Now we can separate the variables:
\[ \frac{dy}{y} = \frac{dx}{a - x} \]

### Step 4: Integrate both sides
Integrating both sides:
\[ \int \frac{dy}{y} = \int \frac{dx}{a - x} \]
The left side integrates to:
\[ \ln |y| + C_1 \]
The right side integrates to:
\[ -\ln |a - x| + C_2 \]
Thus, we have:
\[ \ln |y| = -\ln |a - x| + C \]
where \( C = C_2 - C_1 \).

### Step 5: Exponentiating both sides
Exponentiating both sides gives:
\[ |y| = \frac{K}{|a - x|} \]
where \( K = e^C \).

### Step 6: Removing absolute values
Assuming \( y \) and \( a - x \) are positive in the region we are interested in, we can drop the absolute values:
\[ y = \frac{K}{a - x} \]

### Step 7: Finding the constant K
To find \( K \), we use the point (2a, a):
\[ a = \frac{K}{a - 2a} \]
This simplifies to:
\[ a = \frac{K}{-a} \]
Thus:
\[ K = -a^2 \]

### Step 8: Final equation of the curve
Substituting back for \( K \):
\[ y = \frac{-a^2}{a - x} \]
This can be rewritten as:
\[ y(a - x) = -a^2 \]
or:
\[ y(a - x) + a^2 = 0 \]

### Conclusion
The equation of the curve is:
\[ y(a - x) + a^2 = 0 \]