For the parabola `y^2 = 4ax`, the ratio of the subtangentto the abscissa is

Options

A. `1:1`
B. `2:1`
C. `x:y`
D. `x^2:y`

Answer

To find the ratio of the subtangent to the abscissa for the parabola given by the equation \( y^2 = 4ax \), we can follow these steps:

### Step 1: Differentiate the equation
We start with the equation of the parabola:
\[ y^2 = 4ax \]
To find the slope of the tangent line, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4ax) \]
Using the chain rule on the left side, we get:
\[ 2y \frac{dy}{dx} = 4a \]

### Step 2: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{4a}{2y} = \frac{2a}{y} \]

### Step 3: Find the subtangent
The length of the subtangent \( T \) at any point on the curve is given by the formula:
\[ T = \frac{y}{\frac{dy}{dx}} \]
Substituting the value of \(\frac{dy}{dx}\) we found:
\[ T = \frac{y}{\frac{2a}{y}} = \frac{y^2}{2a} \]

### Step 4: Substitute \( y^2 \) from the parabola's equation
From the equation of the parabola, we know:
\[ y^2 = 4ax \]
Substituting this into the expression for \( T \):
\[ T = \frac{4ax}{2a} = 2x \]

### Step 5: Find the abscissa
The abscissa (the x-coordinate) at the point of tangency is simply \( x \).

### Step 6: Find the ratio of the subtangent to the abscissa
Now we can find the ratio of the subtangent \( T \) to the abscissa \( x \):
\[ \text{Ratio} = \frac{T}{x} = \frac{2x}{x} = 2 \]

### Conclusion
Thus, the ratio of the subtangent to the abscissa is:
\[ \text{Ratio} = 2:1 \]