A point source is at a distance `35cm` on the optical axos from a spherical mirror having a focal length `25cm`. At what distance measured along the optical axis form the concave mirror should a plane mirror (perpendicualr to principle axis) be placed for the image it forms (due to rays falling on it after reflection from the concave mirror) to coincide with the point source?

Explanation

To solve the problem step by step, we will use the mirror formula and the properties of image formation by mirrors.

### Step 1: Identify the given values
- Distance of the point source from the concave mirror (u) = -35 cm (negative because it is measured against the direction of incident light)
- Focal length of the concave mirror (f) = -25 cm (negative for concave mirrors)

### Step 2: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
where:
- \( f \) = focal length of the mirror
- \( v \) = image distance from the mirror
- \( u \) = object distance from the mirror

Substituting the known values into the formula:
\[ \frac{1}{-25} = \frac{1}{v} + \frac{1}{-35} \]

### Step 3: Rearranging the equation
Rearranging the equation gives:
\[ \frac{1}{v} = \frac{1}{-25} + \frac{1}{35} \]

### Step 4: Finding a common denominator
The common denominator for -25 and 35 is 175. Therefore, we can rewrite the equation:
\[ \frac{1}{v} = \frac{-7}{175} + \frac{5}{175} = \frac{-7 + 5}{175} = \frac{-2}{175} \]

### Step 5: Calculate the image distance (v)
Now, we can find \( v \):
\[ v = \frac{175}{-2} = -87.5 \text{ cm} \]
(The negative sign indicates that the image is formed on the same side as the object, which is typical for a concave mirror.)

### Step 6: Determine the distance between the image and the object
Now, we need to find the distance between the object and the image:
- Distance from the object to the image = \( |u| + |v| = 35 + 87.5 = 122.5 \text{ cm} \)

### Step 7: Positioning the plane mirror
For the image formed by the plane mirror to coincide with the point source, the plane mirror must be placed halfway between the object and the image. Thus:
\[ \text{Distance from the concave mirror to the plane mirror} = \frac{122.5}{2} = 61.25 \text{ cm} \]

### Step 8: Calculate the final position of the plane mirror
Since the image is formed at 87.5 cm from the concave mirror, we can find the position of the plane mirror:
\[ \text{Distance from the concave mirror to the plane mirror} = 87.5 - 61.25 = 26.25 \text{ cm} \]
Thus, the distance from the concave mirror to the plane mirror is:
\[ 35 + 26.25 = 61.25 \text{ cm} \]

### Final Answer
The plane mirror should be placed at a distance of **61.25 cm** from the concave mirror along the optical axis.

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