For the parabola `y^(2) = 16x`, the ratio of the length of the subtangent to the abscissa is

Options

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

Answer

To find the ratio of the length of the subtangent to the abscissa for the parabola \( y^2 = 16x \), we will follow these steps:

### Step 1: Differentiate the equation of the parabola
We start with the given equation of the parabola:
\[ y^2 = 16x \]
To find the slope of the tangent line at any point on the parabola, we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y^2) = \frac{d}{dx}(16x) \]
Using the chain rule on the left side, we get:
\[ 2y \frac{dy}{dx} = 16 \]

### Step 2: Solve for \(\frac{dy}{dx}\)
Now, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{16}{2y} = \frac{8}{y} \]

### Step 3: Find the length of the subtangent
The length of the subtangent (LST) at a point \((x, y)\) is given by the formula:
\[ \text{LST} = \frac{y}{\frac{dy}{dx}} \]
Substituting the expression we found for \(\frac{dy}{dx}\):
\[ \text{LST} = \frac{y}{\frac{8}{y}} = \frac{y^2}{8} \]

### Step 4: Substitute \(y^2\) from the parabola's equation
From the equation of the parabola, we know that \(y^2 = 16x\). Therefore, we can substitute this into our expression for LST:
\[ \text{LST} = \frac{16x}{8} = 2x \]

### Step 5: Find the ratio of LST to the abscissa
The abscissa (x-coordinate) is simply \(x\). We need to find the ratio of the length of the subtangent to the abscissa:
\[ \text{Ratio} = \frac{\text{LST}}{x} = \frac{2x}{x} = 2 \]
Thus, the ratio of the length of the subtangent to the abscissa is \(2:1\).

### Final Answer
The ratio of the length of the subtangent to the abscissa is \(2:1\).

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