Question #dafd5

Let at an instant the image distance be #v# for the real object distance #u# during movement of a point object towards a convex mirror of focal length #F# along its axis.

We know that the cojugate foci relation of spherical mirror is as follows.

#color(blue)(1/v+1/u=1/F......[1])#

Imposing sign convention for convex mirror
#F->+ve and u=-u" for real object"# we get equation [1] as

#color(green)(1/v-1/u=1/F.....[2])#

#color(green)(=>1/v=1/u+1/F)#

#color(green)(=>1/v=(u+F)/(uF))#

#color(green)(=>v/u=(F)/(u+F))#

#color(green)(=>v/u=(F)/(u+F)<1.....[3])#

Now differentiating equation [2] w.r to #t# we get

#color(violet)(-1/v^2(dv)/(dt)+1/u^2(du)/(dt)=0.....[4])#

#color(violet)(=>1/v^2(dv)/(dt)=1/u^2(du)/(dt))#

Now #(dv)/(dt)=v_i="speed of image"#

And #(du)/(dt)=v_o="speed of object"#

So equation [4] becomes

#color(red)(1/v^2xxv_i=1/u^2xxv_o)#

#color(red)(=>v_i/v_o=v^2/u^2.....[5])#

Utilising equation [3] and [5] we can say

#color(red)(v_i/v_o=F^2/(u+F)^2)#

So #color(red)(v_i < v_o)" when "abs(u)< abs(F) or abs(u)>abs(F)#

And #color(red)(v_i =v_o)" when "abs(u)< abs(F) and abs(u)->0#

Since #lim_(u->0)(F^2/(u+F)^2)=1#

So we can support option (1) and (3) to be correct