Question

If the distance between an object and its two times magnified virtual image produced by a curved mirror is 15 cm, the focal length of the mirror must be:

• A. \( \frac{10}{3} \) cm
• B. \( -12 \) cm
• C. \( -10 \) cm
• D. \( 15 \) cm

Hint:

For a concave mirror, a magnified virtual image is always formed when the object is placed between the focal point and the mirror.

Answer 1

The Correct Option is C

Step 1: {Understand magnification}
The magnification formula for a mirror is:\[m = -\frac{v}{u}\]Given that the image is virtual and magnified two times:\[m = 2\]Therefore:\[2 = -\frac{v}{u}\]Step 2: {Apply object-image distance}
The total distance between the object and image is:\[u + v = 15\]Substituting \( v = -2u \):\[u + (-2u) = 15\]\[- u = 15 \Rightarrow u = 5 { cm}\]\[v = -2(5) = -10 { cm}\]Step 3: {Calculate focal length using mirror formula}
\[\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\]\[= \frac{1}{-10} + \frac{1}{5} = -\frac{1}{10}\]\[f = -10 { cm}\]The focal length is \( -10 \) cm.