Question #21512

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

Expert's answer

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.

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.

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