A is a point object in a circular track. A light ray starting from the object A is reflected twice by the circular track and returns again to A. Angle of incidence is `alpha` . The distance of A from the center of the circular track is x and the diameter of the circular track from A intersects the path of the ray at a point D whose distance from the center of the circular track is y.
Show that, `tan alpha=sqrt((x-y)/(x+y))`

Explanation

Refer to Fig.1.47.
In `DeltaOBC, OB = OC , "so", angleOBC=angleOBC=alpha`
Also `angleABC = angleACB = 2alpha, "so", AB = AC`
Since `DeltaABC` is isosceles, hence median `AD bot BC`
`therefore " " tanalpha = (y)/(BD) and tan2alpha=(x+y)/(BD)`
`or, " " (tan2alpha)/(tanalpha)=(x+y)/(y) or, (2tanalpha)/(tanalpha(1-tan^(2)alpha))=(x+y)/(y)`
`or, " " 1-tan^(2)alpha=(2y)/(x+y)`
`or, " " tan^(2)alpha=1-(2y)/(x+y)=(x-y)/(x+y) or,tanalpha=sqrt((x-y)/(x+y))`