For the parabola `y^(2)=4ax,` the ratio of the subtangent to the abscissa, is

Options

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

Answer

To solve the problem of finding the ratio of the subtangent to the abscissa for the parabola \( y^2 = 4ax \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given parabola**: The equation of the parabola is given as \( y^2 = 4ax \).

2. **Find the derivative**: To find the slope of the tangent line at any point on the parabola, we need to differentiate the equation with respect to \( x \).
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(4ax) \]
This gives us:
\[ 2y \frac{dy}{dx} = 4a \]
Rearranging this, we find:
\[ \frac{dy}{dx} = \frac{4a}{2y} = \frac{2a}{y} \]

3. **Calculate the length of the subtangent**: The length of the subtangent \( T \) at a point on the curve is given by the formula:
\[ T = \frac{y}{\frac{dy}{dx}} \]
Substituting the value of \( \frac{dy}{dx} \):
\[ T = \frac{y}{\frac{2a}{y}} = \frac{y^2}{2a} \]

4. **Substitute \( y^2 \)**: From the equation of the parabola \( y^2 = 4ax \), we can substitute \( y^2 \) in the expression for \( T \):
\[ T = \frac{4ax}{2a} = 2x \]

5. **Find the abscissa**: The abscissa at the point is simply \( x \).

6. **Calculate the ratio of subtangent to abscissa**: The ratio \( R \) of the subtangent to the abscissa is:
\[ R = \frac{T}{x} = \frac{2x}{x} = 2 \]

### Final Answer:
The ratio of the subtangent to the abscissa is \( 2 \).