Question

A convex mirror forms an image one-fourth the size of the object. If object is at a distance of 0.5 m from mirror the focal length of the mirror is
A. 0.16 m
B. -1.5 m
C. 0.4 m
D. -0.4 m

Answer 1

Apply sign convention properly and use the mirror and magnification formula:

\dfrac{1}{f} = \dfrac{1}{v} + \dfrac{1}{u} \\\\$$ Remember that a convex mirror always produces a virtual-erect image so the magnification will be positive. **Complete step by step answer:** According to the question, size of image = ¼th the size of object and $u=0.5 m$ . Standard sign convection is used here. Object is placed at -0.5 metre from the mirror on the left side. The image will be formed on the right side. If the object is erect then size is positive and vice versa. From the magnification formula, $$m = \dfrac{{{h_i}}}{{{h_o}}} = - \dfrac{v}{u} \\\ \\\ $$ Substituting the values, we have, $$\dfrac{{{h_i}}}{{{h_o}}} = \dfrac{{\dfrac{1}{4}{h_o}}}{{{h_o}}} = - \dfrac{v}{u} \\\ \Rightarrow v = - \dfrac{1}{4}u = - \dfrac{1}{4}( - 0.5) \\\ \Rightarrow v = \dfrac{{0.5}}{4} \\\ $$ Now, substituting the values of v and u in the mirror formula, $$\dfrac{1}{f} = \dfrac{4}{{0.5}} + \dfrac{1}{{( - 0.5)}} = \dfrac{4}{{0.5}} - \dfrac{1}{{0.5}} \\\ \Rightarrow \dfrac{1}{f} = \dfrac{3}{{0.5}} \\\ \Rightarrow f = \dfrac{{0.5}}{3}m = 0.16m \\\ $$ The focal length of the convex mirror is 0.16 m. **So, the correct answer is “Option A”.** **Additional Information:** Even if you forget the correct formula for magnification, whether there was a minus sign or not etc. Always try to remember what type of image the optical instrument produces. Is it virtual-erect or real-inverted? Based on this you can find that for virtual-erect m is always positive and for real-inverted it is negative and hence can find out the correct formula. Also, even after solving, check it based on theory. Here, the convex mirror has virtual focus on the right side of the mirror which is positive. And we got a positive focus hence correct. **Note:** Mostly these types of questions have similar answers in their options. The only difference is in their signs. Therefore, carefully use sign convection otherwise you may end up with correct values but wrong signs.