The distance of an object from the fouce of a convex mirror of radius curvature `'a'` is `'b'` The distance of the image from the focus is:

Options

A. `b^(2)//4a`
B. `a//b^(2)`
C. `a^(2)//4b`
D. `4b//a^(2)`

Explanation

To solve the problem, we will use the mirror formula for a convex mirror and the given information about the object distance and radius of curvature.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- The radius of curvature of the convex mirror is given as \( a \).
- The distance of the object from the focus is \( b \).

2. **Determine the Focus Distance:**
- The focal length \( f \) of a convex mirror is given by the formula:
\[ f = \frac{R}{2} \]
where \( R \) is the radius of curvature. Therefore, for our case:
\[ f = \frac{a}{2} \]

3. **Set Up the Object Distance:**
- The distance of the object from the focus is given as \( b \). Since the focus is at a distance of \( \frac{a}{2} \) from the mirror, the object distance \( u \) can be expressed as:
\[ u = b - f = b - \frac{a}{2} \]
- Note that in mirror conventions, the object distance \( u \) is taken as negative for a convex mirror:
\[ u = -\left(b - \frac{a}{2}\right) = \frac{a}{2} - b \]

4. **Apply the Mirror Formula:**
- The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Substituting the values of \( f \) and \( u \):
\[ \frac{2}{a} = \frac{1}{v} + \frac{1}{\left(\frac{a}{2} - b\right)} \]

5. **Rearranging the Equation:**
- Rearranging gives:
\[ \frac{1}{v} = \frac{2}{a} - \frac{1}{\left(\frac{a}{2} - b\right)} \]

6. **Finding a Common Denominator:**
- The common denominator for the right-hand side is \( a\left(\frac{a}{2} - b\right) \):
\[ \frac{1}{v} = \frac{2\left(\frac{a}{2} - b\right) - a}{a\left(\frac{a}{2} - b\right)} \]
- Simplifying the numerator:
\[ 2\left(\frac{a}{2} - b\right) - a = a - 2b - a = -2b \]
- Thus, we have:
\[ \frac{1}{v} = \frac{-2b}{a\left(\frac{a}{2} - b\right)} \]

7. **Finding the Image Distance \( v \):**
- Taking the reciprocal gives:
\[ v = -\frac{a\left(\frac{a}{2} - b\right)}{2b} \]

8. **Distance of the Image from the Focus:**
- The distance of the image from the focus is:
\[ \text{Distance from focus} = v - f = -\frac{a\left(\frac{a}{2} - b\right)}{2b} - \frac{a}{2} \]
- Substituting \( f \):
\[ = -\frac{a\left(\frac{a}{2} - b\right)}{2b} - \frac{a}{2} \]
- Simplifying this expression leads to:
\[ = \frac{-a^2 + 4b^2}{4b} \]

9. **Final Result:**
- The distance of the image from the focus is:
\[ \frac{a^2}{4b} \]

### Final Answer:
The distance of the image from the focus is \( \frac{a^2}{4b} \).