A particle projected atv a definite angle `prop` to the horizontal passes through points `(a,b) and (b,a)`, referred to horizontal and vertical axes through the points of projection. Show that :
(a) The horizontal range `R = (a^2 + ab + b^2)/(a + b)` .
(b) The angle of projection `prop` is given by
`tan^-1 [(a^2 + ab + b^2)/(ab)]`.

Explanation

(a) The equation of the trajectory of the particle
`y = x tan prop (1 - (x)/( R))`….(i)
Since the particle passes through the points with coordinates `(a, b) and (b,a)`, they should satisfy the equation of the curve.
`b = a tan prop (1 - (a)/(R )) and (ii) a = b tan prop (1 - (b)/( R))` ...(ii)
Dividing `(b^2)/(a^2) = ((1 - (a)/(R)))/((1 - (b)/(R)))`
`rArr b^2 - (b^3)/(R) = a^2 - (a^3)/(R)`
`rArr (1)/(R) [a^3 - b^3] = a^2 -b^2`
`rArr R = (a^3 - b^3)/(a^2 - b^2) = ((a- b)(a^2 + ab + b^2))/((a - b)(a + b)) = (a^2 + ab + b^2)/(a + b)`
Hence, proved. `[:. a != b]`
(b) Substituting the expression for `R` in (ii),
`(b)/(a) = tan prop [1 - (a( + b))/(a^2 + ab + b^2)]`
=`tan prop [(a^2 + ab + b^2 - a^2 - ab)/(a^2 + ab + b^2)]`
`tan prop = (a^2 + ab + b^2)/(ab) rArr prop = tan^-1 [(a^2 + ab + b^2)/(ab)]`
Hence proved.