From the focus of the parabola `y^2=2px` as centre, a circle id drawn so that a common chord of the curve is equidistant from the vertex and the focus. The radius of the circle is

Options

A. `(p)/(4)`
B. `(3p)/(4)`
C. `(3p)/(8)`
D. `(sqrt(3)p)/(8)`

Answer

To solve the problem, we need to find the radius of a circle drawn from the focus of the parabola \( y^2 = 2px \) such that a common chord of the circle and the parabola is equidistant from the vertex and the focus.

### Step-by-Step Solution:

1. **Identify the Focus and Vertex of the Parabola:**
The given parabola is \( y^2 = 2px \). From the standard form \( y^2 = 4ax \), we can identify that:
- \( 4a = 2p \) implies \( a = \frac{p}{2} \).
- The focus \( F \) of the parabola is at \( \left(\frac{p}{2}, 0\right) \).
- The vertex \( V \) of the parabola is at the origin \( (0, 0) \).

2. **Determine the Midpoint of the Segment between the Focus and the Vertex:**
The midpoint \( P \) of the segment joining the focus \( F \) and the vertex \( V \) can be calculated using the midpoint formula:
\[ P = \left( \frac{0 + \frac{p}{2}}{2}, \frac{0 + 0}{2} \right) = \left( \frac{p}{4}, 0 \right). \]

3. **Identify the Common Chord:**
The common chord of the circle and the parabola must be perpendicular to the line segment \( VF \) and pass through point \( P \). Since the x-coordinate of \( P \) is \( \frac{p}{4} \), the common chord will be a vertical line at \( x = \frac{p}{4} \).

4. **Find the Points of Intersection of the Chord with the Parabola:**
Substitute \( x = \frac{p}{4} \) into the parabola's equation:
\[ y^2 = 2p \cdot \frac{p}{4} = \frac{p^2}{2}. \]
Thus, the y-coordinates of the intersection points are:
\[ y = \pm \frac{p}{\sqrt{2}}. \]

5. **Equation of the Circle:**
The circle is centered at the focus \( F \left(\frac{p}{2}, 0\right) \) with radius \( r \). The general equation of the circle is:
\[ \left(x - \frac{p}{2}\right)^2 + y^2 = r^2. \]

6. **Substituting the Points of Intersection into the Circle's Equation:**
We substitute the point \( P \left(\frac{p}{4}, \frac{p}{\sqrt{2}}\right) \) into the circle's equation to find \( r \):
\[ \left(\frac{p}{4} - \frac{p}{2}\right)^2 + \left(\frac{p}{\sqrt{2}}\right)^2 = r^2. \]
Simplifying this:
\[ \left(-\frac{p}{4}\right)^2 + \frac{p^2}{2} = r^2. \]
\[ \frac{p^2}{16} + \frac{p^2}{2} = r^2. \]
To combine these, we need a common denominator:
\[ \frac{p^2}{16} + \frac{8p^2}{16} = r^2 \implies \frac{9p^2}{16} = r^2. \]

7. **Finding the Radius:**
Taking the square root gives:
\[ r = \sqrt{\frac{9p^2}{16}} = \frac{3p}{4}. \]

### Final Answer:
The radius of the circle is \( \frac{3p}{4} \).