Two parallel straight lines are inclined to the horizon at an angle `prop`. A particle is projected from a point mid way between them so as to graze one of the lines and strikes the other at right angle. Show that if `theta` is the angle between the direction of projection and either of lines, then `tan theta = (sqrt(2 - 1)) cot prop`.

Explanation

A to B : `0^2 = u^2 sin^2 - 2 (g cos prop) h`
A to C : -h `u sin theta t - (1)/(2) g cos prop t^2`
And `0 = u cos theta - g sin prop t`
`rArr t = (u cos theta)/(g sin prop) -h =(u^2 sin theta cos theta)/(g sin prop) -(1)/(2) (g cos prop u^2 cos^2 theta)/(g^2 sin^2 prop)`
`(-u^2 sin^2 theta)/(2 g cos prop) =(u^2 sin theta cos theta)/(g sin prop) -(u^2 cos prop cos^2 theta)/(2 g sin^2 prop)`
`(-sin^2 theta)/(2 cos prop) =(sin theta cos theta)/(sin prop) -(cos prop cos^2 theta)/(2 sin^2 prop)`
Multiplying both sides by `(2 cos prop)/(cos^2 theta)`, we get
`-tan^2 theta = 2 cot prop tan theta - cot^2 prop = 0`
`tan^2 theta + 2 cot prop tan theta - cot^2 prop = 0`
`tan theta = (-2 cot prop + sqrt(4 cot^2 prop + 4 cot^2 prop))/(2)`
=`(sqrt(2) -1) cot prop`.
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