A circle is drawn to pass through the extremities of the latus rectum of the parabola `y^2=8xdot` It is given that this circle also touches the directrix of the parabola. Find the radius of this circle.

Explanation

To solve the problem step by step, we will find the radius of the circle that passes through the extremities of the latus rectum of the parabola \( y^2 = 8x \) and also touches the directrix of the parabola.

### Step 1: Identify the parameters of the parabola
The given parabola is \( y^2 = 8x \). We can rewrite it in the standard form \( y^2 = 4ax \), where \( 4a = 8 \).

**Calculation:**
\[ 4a = 8 \implies a = 2 \]

### Step 2: Find the focus and directrix of the parabola
For the parabola \( y^2 = 4ax \):
- The focus is at \( (a, 0) = (2, 0) \).
- The directrix is the line \( x = -a = -2 \).

### Step 3: Find the extremities of the latus rectum
The latus rectum of a parabola is a line segment perpendicular to the axis of symmetry that passes through the focus. The length of the latus rectum is \( 4a \).

**Calculation:**
\[ \text{Length of latus rectum} = 4a = 8 \]
The extremities of the latus rectum can be found at:
\[ \left( a, \pm 2a \right) = \left( 2, \pm 4 \right) \]
Thus, the points are \( (2, 4) \) and \( (2, -4) \).

### Step 4: Determine the diameter of the circle
The diameter of the circle that passes through the points \( (2, 4) \) and \( (2, -4) \) is the distance between these two points.

**Calculation:**
\[ \text{Diameter} = \text{Distance between } (2, 4) \text{ and } (2, -4) = 4 - (-4) = 8 \]

### Step 5: Find the radius of the circle
The radius \( r \) of the circle is half of the diameter.

**Calculation:**
\[ r = \frac{\text{Diameter}}{2} = \frac{8}{2} = 4 \]

### Step 6: Verify that the circle touches the directrix
Since the circle is drawn considering the latus rectum as the diameter, it will touch the directrix \( x = -2 \). The center of the circle is at \( (2, 0) \) and the radius is \( 4 \). The distance from the center to the directrix is:
\[ \text{Distance} = 2 - (-2) = 4 \]
This confirms that the circle touches the directrix.

### Final Answer
The radius of the circle is \( \boxed{4} \).

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