A particle is projectedfrom origin at an angle is selectedas X-axis and verticle as Y-axis . It is found that particle passes throughthe points (a, b) and (b, a). Calculate range and angle of projection .

A particle is projectedfrom origin at an angle is selectedas X-axis an

1.	A particle is projectedfrom origin at an angle is selectedas X-axis and verticle as Y-axis . It is found that particle passes throughthe points (a, b) and (b, a). Calculate range and angle of projection .
Answer» Solution :Equartion of trajectory in terms of range of PROJECTILE can be written as follows : `y=x(1-(x)/(R))TAN theta` Given coodinates must satisfy theabove EQUATION. Hence we can write the following : `b=a(1-(a)/(R)) tan theta` `a=b(1-(b)/(R)) tan theta` Dividing the abovetwo EQUATIONS, we can write the following: `(b^(2))/(a^(2))=(1-(a)/(R))/(1-(b)/(R))` `rArr (b^(2))/(a^(2))=(R-a)/(R-b)rArrb^(2)R-b^(3)=a^(2)R-a^(3)` `rArr R=(a^(3)-b^(3))/(a^(2)-b^(2))=((a-b)(a^(2)_ab+b^(2)))/((a-b)(a+b))` `rArr R=(a^(2)+ab+b^(2))/(a+b)` Now we can substitute R in equation (i) to getthe angle ofprojection . `b=a(1-((a)/(a^(2)+ab+b^(2)))/(a+b))tan theta` `b=a(1-(a^(2)+ab)/(a^(2)+ab+b^(2)))tan theta` `rArr (b)/(c) ((a^(2)+ab+b^(2)0a^(2)-ab)/(a^(2)+ab+b^(2))) tan theta` `rArr (b)/(a)=((b^(2))/(a^(2)+ab+b^(2))) tan theta` `rArr theta =(a^(2)+ab+b^(2))/(ab)` `rArr theta =tan ^(-1)((a^(2)+ab+b^(2))/(ab))`.