A virtual image three times the size of the object is obtained with a concave mirror of radius of curvature 36cm. The distance of the object from the mirror is

Options

A. 5 cm
B. 12 cm
C. 10 cm
D. 20 cm

Answer

To solve the problem, we need to find the distance of the object from a concave mirror given that the virtual image formed is three times the size of the object. We also know the radius of curvature of the mirror.

### Step-by-Step Solution:

**Step 1: Understand the given values.**
- The magnification (m) of the image is given as 3 (since the image is virtual and upright).
- The radius of curvature (R) of the concave mirror is 36 cm.

**Hint:** Recall that the magnification for mirrors is defined as the ratio of the height of the image to the height of the object.

**Step 2: Calculate the focal length (f) of the mirror.**
- The focal length (f) is related to the radius of curvature (R) by the formula:
\[ f = \frac{R}{2} \]
- Substituting the value of R:
\[ f = \frac{36 \, \text{cm}}{2} = 18 \, \text{cm} \]

**Hint:** Remember that for a concave mirror, the focal length is negative.

**Step 3: Apply the magnification formula.**
- The magnification (m) is also given by the formula:
\[ m = -\frac{b}{u} \]
where \( b \) is the image distance and \( u \) is the object distance.
- Since the image is virtual and upright, we take \( m = 3 \):
\[ 3 = -\frac{b}{u} \]
This implies:
\[ b = -3u \]

**Hint:** Be cautious with the signs; virtual images formed by concave mirrors have negative image distances.

**Step 4: Use the mirror formula.**
- The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{b} \]
- Substituting the values of \( f \) and \( b \):
\[ \frac{1}{-18} = \frac{1}{u} + \frac{1}{-3u} \]

**Hint:** Substitute the expression for \( b \) from the previous step into the mirror formula.

**Step 5: Simplify the equation.**
- The equation becomes:
\[ \frac{1}{-18} = \frac{1}{u} - \frac{1}{3u} \]
- Finding a common denominator for the right side:
\[ \frac{1}{u} - \frac{1}{3u} = \frac{3 - 1}{3u} = \frac{2}{3u} \]
- Thus, we have:
\[ \frac{1}{-18} = \frac{2}{3u} \]

**Hint:** Cross-multiply to solve for \( u \).

**Step 6: Solve for \( u \).**
- Cross-multiplying gives:
\[ 3u = -36 \quad \Rightarrow \quad u = -12 \, \text{cm} \]

**Step 7: Interpret the result.**
- The negative sign indicates that the object is placed in front of the mirror, which is consistent with the convention for concave mirrors.

### Final Answer:
The distance of the object from the mirror is **12 cm** (in front of the mirror).