$A$ student focused the image of a candle flame on a white screen using a convex lens. He noted down the positions of the candle,screen,and lens as follows:
Position of candle $= 12.0 \, cm$
Position of convex lens $= 50.0 \, cm$
Position of the screen $= 88.0 \, cm$
$(i)$ What is the focal length of the convex lens?
$(ii)$ Where will the image be formed if he shifts the candle towards the lens to a position of $31.0 \, cm$?

(A) $(i)$ The object distance $u = 50.0 - 12.0 = 38.0 \, cm$. The image distance $v = 88.0 - 50.0 = 38.0 \, cm$. Since $u = v$,the object is placed at $2f$. Thus,$2f = 38.0 \, cm$,which gives the focal length $f = 19.0 \, cm$.
$(ii)$ If the candle is shifted to $31.0 \, cm$,the new object distance $u' = 50.0 - 31.0 = 19.0 \, cm$. Since the object is now placed at the focus $(u' = f = 19.0 \, cm)$,the image will be formed at infinity.