To solve this problem, let’s go step-by-step through the physics principles involved, using the given information about the concave mirror.
### Given:
- Focal length, \(\( f = 51 \)\) cm
- The image is inverted and 4 times smaller than the object, implying the magnification, \(\( m = -\frac{1}{4} \)\)
### Step 1: Use the Magnification Formula
The magnification formula for mirrors is given by:
\(\[ m = -\frac{v}{u} \]\)
Where:
- \(\( v \)\) is the image distance
- \(\( u \)\) is the object distance
Given \(\( m = -\frac{1}{4} \)\), we can rearrange this formula to express \(\( v \)\) in terms of \(\( u \)\):
\(\[ -\frac{1}{4} = -\frac{v}{u} \Rightarrow v = \frac{u}{4} \]\)
### Step 2: Use the Mirror Formula
The mirror formula is given by:
\(\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]\)
### Step 3: Substitute \(\( v \)\) into the Mirror Formula
From the magnification formula, we have \(\( v = \frac{u}{4} \)\). Substitute this into the mirror equation:
\(\[ \frac{1}{51} = \frac{1}{\frac{u}{4}} + \frac{1}{u} \]\)
\(\[ \frac{1}{51} = \frac{4}{u} + \frac{1}{u} \]\)
\(\[ \frac{1}{51} = \frac{4 + 1}{u} \]\)
\(\[ \frac{1}{51} = \frac{5}{u} \]\)
### Step 4: Solve for \(\( u \)\)
Rearranging the equation to solve for \(\( u \)\) gives:
\(\[ u = 51 \times 5 \]\)
\(\[ u = 255 \]\)
So, the object distance \(\( u \)\) is 255 cm.
### Step 5: Calculate the Image Distance \(\( v \)\)
Now, we use \(\( v = \frac{u}{4} \)\) to find the image distance:
\(\[ v = \frac{255}{4} \]\)
\(\[ v = 63.75 \]\)
So, the image distance \(\( v \)\) is 63.75 cm.
### Conclusion:
The position of the object is 255 cm from the mirror, and the position of the resulting image is 63.75 cm from the mirror.
