If the point P(4, -2) is the one end of the focal chord PQ of the parabola `y^(2)=x,` then the slope of the tangent at Q, is

Explanation

To find the slope of the tangent at point Q of the parabola \( y^2 = x \), given that point P(4, -2) is one end of the focal chord PQ, we can follow these steps:

### Step 1: Understand the Parabola
The equation of the parabola is given as \( y^2 = x \). This can also be expressed in the standard form where the vertex is at the origin (0,0) and opens to the right.

### Step 2: Identify the Focal Chord
A focal chord of a parabola is a line segment that passes through the focus of the parabola. For the parabola \( y^2 = x \), the focus is at the point \( (1/4, 0) \).

### Step 3: Use the Properties of Focal Chords
For any point \( P(t_1) \) on the parabola, the coordinates can be expressed as:
\[ P(t_1) = (t_1^2, 2t_1) \]
Given \( P(4, -2) \), we can equate:
\[ t_1^2 = 4 \quad \text{and} \quad 2t_1 = -2 \]
From \( 2t_1 = -2 \), we find \( t_1 = -1 \).

### Step 4: Find the Corresponding \( t_2 \) for Point Q
For the focal chord, the product of the parameters \( t_1 \) and \( t_2 \) is -1:
\[ t_1 \cdot t_2 = -1 \implies -1 \cdot t_2 = -1 \implies t_2 = 1 \]
Thus, the coordinates of point Q are:
\[ Q(t_2) = (t_2^2, 2t_2) = (1^2, 2 \cdot 1) = (1, 2) \]

### Step 5: Find the Slope of the Tangent at Point Q
To find the slope of the tangent line at point Q, we differentiate the equation of the parabola:
\[ y^2 = x \implies 2y \frac{dy}{dx} = 1 \implies \frac{dy}{dx} = \frac{1}{2y} \]
At point \( Q(1, 2) \):
\[ \frac{dy}{dx} = \frac{1}{2 \cdot 2} = \frac{1}{4} \]

### Conclusion
The slope of the tangent at point Q is \( \frac{1}{4} \).