two straight line u=0 and v=0 passes through the origin and the angle between them is tan inverse (7/4). If the ratio of the slope of v=0 and u=0 is 9/2 then their equations are
Let A be the slope of line u and B be the slope of v, sotheir equations are the following: u: y=Ax v: y=Bx Let alpha and beta be the angles between the positivedirection of x-axis and lines u and v respectively. Then tan(alpha) = A tan(beta) = B By assumption B/A = 9/2, so beta>alpha, and also beta-alpha =tan_inverse(7/4), so tan(beta-alpha)= 7/4 On the other hand it by well-known formula
tan(beta-alpha) = [ tan(beta)-tan(alpha) ] / [ 1 +tan(beta)*tan(alpha) ] = =(B-A)/(1+BA) So we get the following system of equations: B/A = 9/2 (B-A)/(1+BA) =7/4 From the first one we obtain B = 9A/2 = 4.5A Substituting into the second equation we get B-A = 7(1+AB)/4 4.5A - A = 7(1+ 4.5A^2)/4 14A = 7(1 +4.5A^2) 2A = 1 + 4.5A^2 4.5 A^2 - 2A +1 = 0
9 A^2 - 4A + 2= 0 D = 4^4 - 4*9 =16-36=-20<0 Thus the discriminant of this equation is negative, andtherefore there are no solution. Thus the situation described in the problem isimpossible.