A converging beam of light rays in incident on a concave spherical mirror whose radius of curvature is `0.8m`. Determine the position of the point on the optical axis of the mirror where the reflected rays intersect, if the extensions of the incident rays intersect the optical axis `40cm` form the mirror's pole.

Explanation

To solve the problem step by step, we will use the mirror formula and the information provided in the question.

### Step 1: Understand the given data
- Radius of curvature (R) = 0.8 m = 80 cm (since we need to work in centimeters)
- The distance of the incident rays' intersection point from the mirror's pole (u) = 40 cm (this is positive as per the sign convention for object distance)
- The focal length (F) can be calculated using the formula \( F = \frac{R}{2} \).

### Step 2: Calculate the focal length (F)
Using the formula for focal length:
\[ F = \frac{R}{2} = \frac{80 \, \text{cm}}{2} = 40 \, \text{cm} \]
Since the mirror is concave, the focal length is negative:
\[ F = -40 \, \text{cm} \]

### Step 3: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- \( f \) = focal length
- \( v \) = image distance (what we need to find)
- \( u \) = object distance

### Step 4: Substitute the known values into the mirror formula
Substituting the values we have:
\[ \frac{1}{-40} = \frac{1}{v} + \frac{1}{40} \]

### Step 5: Rearranging the equation
To find \( \frac{1}{v} \), we can rearrange the equation:
\[ \frac{1}{v} = \frac{1}{-40} - \frac{1}{40} \]
Finding a common denominator (which is 40):
\[ \frac{1}{v} = -\frac{1}{40} - \frac{1}{40} = -\frac{2}{40} = -\frac{1}{20} \]

### Step 6: Calculate the image distance (v)
Now, taking the reciprocal to find \( v \):
\[ v = -20 \, \text{cm} \]

### Conclusion
The position of the point on the optical axis of the mirror where the reflected rays intersect is at \( -20 \, \text{cm} \) from the mirror's pole.