A concave mirror and a convex mirror are placed coaxially face to face. The focal length of each of them is f and distance between them is 4f. A point source is so placed on their common axis in between the two mirrors that if the first reflction is considered to take place on the convex mirror, the final coincides with teh point source. Determine the position of the source.

Explanation

Let the point source O be situated at a distance x from the convex mirror [Fig.1.48].

If the image distance is `v_(1)` then,
`(1)/(v_(1))+(1)/(-x)=(1)/(f) or, (1)/(v_(1))=+(1)/(f)+(1)/(x)=((x+f))/(fx) or, v_(1)=(fx)/(x+f)`
This image will be formed at `O_(1)` behind the convex mirror. Now this image will act as the object of the concave mirror.
`therefore` Object distance for the concave mirror,
`u_(2)=4f+v_(1)=4f+(fx)/(x+f)=(5fx+4f^(2))/(x+f)`
`therefore` Image distance, `v_(2) = 4f - x`
[`therefore` the final image coincides with the object]
`therefore " " (1)/(v_(2))+(1)/(u_(2))=(1)/(f) or, (1)/(4f-x)+(x+f)/(5fx+4f^(2))=(4x-3f)/(f(5x+4f))`
`or, " " (1)/(4f-x)=(1)/(f)-(x+f)/(5fx+4f^(2))=(5x+4f-x-f)/(f(5x+4f))=(4x+3f)/(f(5x+4f))`
`or, " " x^(2)-2fx-2f^(2)=0`
`or, " " x=(2f pm sqrt(4f^(2)+8f^(2)))/(2)=f pm fsqrt3`
`therefore " " x=f(1+sqrt3)` [neglecting the negative value]
`therefore` The distance of the source from convex mirror `=f(1+sqrt3).`