The radius of curvature of a convex spherical mirror is `1.2m`. How far away from the mirror is an object of height `1.2m` if the distance between its virtual image and the mirror is `0.35m`? What is the height of the image?[Apply formula for paraxial rays].

Explanation

To solve the problem step by step, we will use the mirror formula and the magnification formula for mirrors.

### Step 1: Identify the given data
- Radius of curvature (R) = 1.2 m = 120 cm (since we will convert everything to centimeters)
- Distance of the virtual image from the mirror (v) = -0.35 m = -35 cm (the negative sign indicates that the image is virtual)
- Height of the object (h_o) = 1.2 m = 120 cm

### Step 2: Calculate the focal length (f)
The focal length (f) of a spherical mirror is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the value of R:
\[ f = \frac{120 \text{ cm}}{2} = 60 \text{ cm} \]
Since it is a convex mirror, the focal length is negative:
\[ f = -60 \text{ cm} \]

### Step 3: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting the known values:
\[ \frac{1}{-60} = \frac{1}{-35} + \frac{1}{u} \]

### Step 4: Solve for \( \frac{1}{u} \)
Rearranging the equation:
\[ \frac{1}{u} = \frac{1}{-60} - \frac{1}{-35} \]
Finding a common denominator (which is 420):
\[ \frac{1}{u} = \frac{-7}{420} + \frac{12}{420} \]
\[ \frac{1}{u} = \frac{5}{420} \]
Thus:
\[ u = \frac{420}{5} = 84 \text{ cm} \]

### Step 5: Calculate the height of the image (h_i)
The magnification (m) of the mirror is given by:
\[ m = \frac{h_i}{h_o} = -\frac{v}{u} \]
Substituting the known values:
\[ m = -\frac{-35}{84} = \frac{35}{84} \]
Now, using the magnification to find the height of the image:
\[ h_i = m \cdot h_o \]
\[ h_i = \frac{35}{84} \cdot 120 \]
Calculating:
\[ h_i = \frac{35 \cdot 120}{84} = \frac{4200}{84} = 50 \text{ cm} \]

### Final Answers
- The distance of the object from the mirror (u) = 84 cm
- The height of the image (h_i) = 50 cm