Answer to Question #21512 in Geometry for sachin

Question

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

Answer 1

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.