A circle of radius 4 drawn on a chord of the parabola `y^(2)=8x` as diameter touches the axis of the parabola. Then the slope of the chord is

Options

A. `(1)/(2)`
B. `(3)/(4)`
C. 1
D. 2

Answer

To find the slope of the chord of the parabola \( y^2 = 8x \) that serves as the diameter of a circle with radius 4, we can follow these steps:

### Step 1: Identify Points on the Parabola
Let the points \( A \) and \( B \) on the parabola be represented as:
- \( A \left( \frac{y_1^2}{8}, y_1 \right) \)
- \( B \left( \frac{y_2^2}{8}, y_2 \right) \)

### Step 2: Find the Center of the Circle
The center \( O \) of the circle, which is the midpoint of the chord \( AB \), can be calculated as follows:
\[ O = \left( \frac{\frac{y_1^2}{8} + \frac{y_2^2}{8}}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{y_1^2 + y_2^2}{16}, \frac{y_1 + y_2}{2} \right) \]

### Step 3: Use the Circle's Radius
Since the circle has a radius of 4 and touches the x-axis (the axis of the parabola), the y-coordinate of the center must equal the radius:
\[ \frac{y_1 + y_2}{2} = 4 \]
This implies:
\[ y_1 + y_2 = 8 \]

### Step 4: Calculate the Slope of the Chord
The slope \( m \) of the chord \( AB \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ m = \frac{y_2 - y_1}{\frac{y_2^2}{8} - \frac{y_1^2}{8}} = \frac{y_2 - y_1}{\frac{1}{8}(y_2^2 - y_1^2)} = \frac{8(y_2 - y_1)}{y_2^2 - y_1^2} \]
Using the difference of squares:
\[ y_2^2 - y_1^2 = (y_2 - y_1)(y_2 + y_1) \]
Thus, we can simplify the slope:
\[ m = \frac{8(y_2 - y_1)}{(y_2 - y_1)(y_2 + y_1)} = \frac{8}{y_2 + y_1} \]

### Step 5: Substitute the Sum of y-coordinates
Since \( y_1 + y_2 = 8 \):
\[ m = \frac{8}{8} = 1 \]

### Conclusion
The slope of the chord \( AB \) is:
\[ \boxed{1} \]