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 required abscissa when the ordinate is \( \frac{1}{3} \).

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

### Step 2: Equation of the tangent
Let the point on the curve be \( (x, y) \). The equation of the tangent at this point can be expressed using the point-slope form:
\[ y - y_1 = m(x - x_1) \]
where \( m = \frac{dy}{dx} \) at the point \( (x, y) \).

### Step 3: Finding the x-intercept of the tangent
To find the x-intercept, we set \( y = 0 \):
\[ 0 - y = \frac{dy}{dx}(x - x) \implies -y = \frac{dy}{dx}(x - x) \]
Rearranging gives:
\[ y = \frac{dy}{dx}(x - x_0) \implies x = x_0 + \frac{y}{\frac{dy}{dx}} \]
Let \( x_0 = 2x \) (as per the problem statement), we have:
\[ x = 2x - \frac{y}{\frac{dy}{dx}} \implies \frac{y}{\frac{dy}{dx}} = x \]

### Step 4: Rearranging the equation
From the above, we can rearrange to get:
\[ \frac{dy}{dx} = \frac{y}{x} \]

### Step 5: Separating variables
We can separate the variables:
\[ \frac{dy}{y} = \frac{dx}{x} \]

### Step 6: Integrating both sides
Integrating both sides gives:
\[ \ln |y| = \ln |x| + C \]
Exponentiating both sides leads to:
\[ y = kx \]
where \( k = e^C \).

### Step 7: Finding the constant using the point (1, 2)
Using the point \( (1, 2) \):
\[ 2 = k \cdot 1 \implies k = 2 \]
Thus, the equation of the curve is:
\[ y = 2x \]

### Step 8: Finding the abscissa when ordinate is \( \frac{1}{3} \)
Now, we need to find \( x \) when \( y = \frac{1}{3} \):
\[ \frac{1}{3} = 2x \implies x = \frac{1}{6} \]

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