Distance of a point P on the parabola `y^(2)=48x` from the focus is l and its distance from the tangent at the vertex is d then l-d is equal to

Options

A. 4
B. 8
C. 12
D. 16

Answer

To solve the problem, we need to find the value of \( l - d \) where \( l \) is the distance from a point \( P \) on the parabola \( y^2 = 48x \) to the focus, and \( d \) is the distance from the point \( P \) to the tangent at the vertex.

### Step 1: Identify the parameters of the parabola
The given equation of the parabola is \( y^2 = 48x \). This can be rewritten in the standard form \( y^2 = 4ax \), where \( 4a = 48 \). Thus, we find:
\[ a = \frac{48}{4} = 12 \]

### Step 2: Determine the focus and vertex of the parabola
For the parabola \( y^2 = 4ax \):
- The vertex is at \( (0, 0) \).
- The focus is at \( (a, 0) = (12, 0) \).

### Step 3: Identify the tangent at the vertex
The tangent line at the vertex of the parabola \( y^2 = 4ax \) is the x-axis, which can be expressed as:
\[ y = 0 \]

### Step 4: Determine the distances \( l \) and \( d \)
Let the point \( P \) on the parabola be represented as \( (x_1, y_1) \). Since \( P \) lies on the parabola, it satisfies:
\[ y_1^2 = 48x_1 \]

**Distance \( l \) from point \( P \) to the focus:**
The distance \( l \) from point \( P(x_1, y_1) \) to the focus \( (12, 0) \) is given by:
\[ l = \sqrt{(x_1 - 12)^2 + (y_1 - 0)^2} = \sqrt{(x_1 - 12)^2 + y_1^2} \]
Substituting \( y_1^2 = 48x_1 \):
\[ l = \sqrt{(x_1 - 12)^2 + 48x_1} \]

**Distance \( d \) from point \( P \) to the tangent at the vertex:**
The distance \( d \) from point \( P(x_1, y_1) \) to the x-axis (tangent line \( y = 0 \)) is simply the absolute value of the y-coordinate:
\[ d = |y_1| \]

### Step 5: Calculate \( l - d \)
Now we need to find \( l - d \):
\[ l - d = \sqrt{(x_1 - 12)^2 + 48x_1} - |y_1| \]
Since \( y_1 = \sqrt{48x_1} \) (taking the positive root for simplicity):
\[ d = \sqrt{48x_1} \]
Thus,
\[ l - d = \sqrt{(x_1 - 12)^2 + 48x_1} - \sqrt{48x_1} \]

### Step 6: Substitute \( x_1 \) with \( a \)
From the earlier steps, we know that for any point on the parabola, the distance \( l \) can be expressed in terms of \( a \):
\[ l = a + d \]
Thus,
\[ l - d = a \]
Given \( a = 12 \):
\[ l - d = 12 \]

### Final Answer
Therefore, the value of \( l - d \) is:
\[ \boxed{12} \]