A curve is such that the intercept of the x-axis cut off between the origin and the tangent at a point is twice the abscissa and which passes through the point (1, 2). If the ordinate of the point on the curve is `1/3` then the value of abscissa is :

Answer

To solve the problem step by step, we will derive the equation of the curve based on the given conditions and find the abscissa when the ordinate is \( \frac{1}{3} \).

### Step 1: Understand the problem
We are given that the intercept on the x-axis cut off between the origin and the tangent at a point on the curve is twice the abscissa of that point. The curve also passes through the point (1, 2).

### Step 2: Set up the tangent equation
Let the point on the curve be \( (x, y) \). The slope of the tangent at this point is given by \( \frac{dy}{dx} \). The equation of the tangent line at point \( (x, y) \) can be expressed using the point-slope form:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (x, y) \) and \( m = \frac{dy}{dx} \):
\[ y - y = \frac{dy}{dx}(x - x) \implies y = \frac{dy}{dx}(x - x) + y \]

### Step 3: Find the x-intercept of the tangent
To find the x-intercept, we set \( y = 0 \):
\[ 0 - y = \frac{dy}{dx}(x - x) \implies -y = \frac{dy}{dx}(x - x) \]
Solving for \( x \):
\[ x = x - \frac{y}{\frac{dy}{dx}} \implies x = x + \frac{y}{\frac{dy}{dx}} \]
The x-intercept \( (x_0, 0) \) is given by:
\[ x_0 = x - \frac{y}{\frac{dy}{dx}} \]

### Step 4: Set up the relationship
According to the problem, the x-intercept \( x_0 \) is twice the abscissa \( x \):
\[ x - \frac{y}{\frac{dy}{dx}} = 2x \]
Rearranging gives:
\[ -\frac{y}{\frac{dy}{dx}} = x \implies \frac{dy}{dx} = -\frac{y}{x} \]

### Step 5: Solve the differential equation
This is a separable differential equation:
\[ \frac{dy}{y} = -\frac{dx}{x} \]
Integrating both sides:
\[ \int \frac{dy}{y} = -\int \frac{dx}{x} \implies \ln |y| = -\ln |x| + C \]
Exponentiating both sides:
\[ |y| = \frac{C}{x} \]
Thus, we can write:
\[ y = \frac{k}{x} \]
where \( k \) is a constant.

### Step 6: Use the point (1, 2) to find k
Substituting the point (1, 2) into the equation:
\[ 2 = \frac{k}{1} \implies k = 2 \]
So the equation of the curve is:
\[ y = \frac{2}{x} \]

### Step 7: Find the abscissa when the ordinate is \( \frac{1}{3} \)
Now we need to find \( x \) when \( y = \frac{1}{3} \):
\[ \frac{1}{3} = \frac{2}{x} \]
Cross-multiplying gives:
\[ x = 6 \]

### Final Answer
The value of the abscissa when the ordinate is \( \frac{1}{3} \) is:
\[ \boxed{6} \]