To solve the problem step by step, we will follow these steps:
### Step 1: Identify Given Data
- Distance of the point source \( S \) from the concave mirror \( (u) = -60 \, \text{cm} \) (the negative sign indicates that the object is in front of the mirror).
- Radius of curvature of the concave mirror \( (R) = 80 \, \text{cm} \).
- Focal length \( (F) \) of the concave mirror is given by the formula \( F = \frac{R}{2} = \frac{80}{2} = 40 \, \text{cm} \).
### Step 2: Apply the Mirror Formula
The mirror formula is given by:
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\]
Substituting the known values:
\[
\frac{1}{40} = \frac{1}{v} + \frac{1}{-60}
\]
Rearranging gives:
\[
\frac{1}{v} = \frac{1}{40} + \frac{1}{60}
\]
Finding a common denominator (120):
\[
\frac{1}{v} = \frac{3}{120} + \frac{2}{120} = \frac{5}{120}
\]
Thus,
\[
v = \frac{120}{5} = 24 \, \text{cm}
\]
This means the image is formed at a distance of \( 24 \, \text{cm} \) in front of the mirror (real image).
### Step 3: Determine the Distance for the Flat Mirror
To have the rays converge again at point \( S \), we need to place the flat mirror such that the distance from the flat mirror to the image formed by the concave mirror is equal to the distance from the flat mirror to the point source \( S \).
The image formed by the concave mirror is at \( 24 \, \text{cm} \), and the point source \( S \) is at \( 60 \, \text{cm} \). The distance between the image and the point source is:
\[
60 \, \text{cm} - 24 \, \text{cm} = 36 \, \text{cm}
\]
To place the flat mirror, we can place it halfway between the image and the source:
\[
\text{Distance from the concave mirror to the flat mirror} = 24 \, \text{cm} + \frac{36 \, \text{cm}}{2} = 24 \, \text{cm} + 18 \, \text{cm} = 42 \, \text{cm}
\]
Thus, the flat mirror should be placed at a distance of \( 42 \, \text{cm} \) from the concave mirror.
### Step 4: Analyze the Second Part of the Question
If the rays are first reflected from the flat mirror before reaching the concave mirror, the position where the rays meet will not change. The flat mirror will create a virtual image at \( 60 \, \text{cm} \) (the same as the original source position), and the concave mirror will reflect it back to the same point \( S \).
### Final Answer
The flat mirror should be placed at a distance of **42 cm** from the concave mirror. The position of the point where the rays meet will **not change** if they are first reflected from the flat mirror.
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