SOLUTION: The acute angle between the line x-2y+3=0 and the line y=mx is 45 degrees. Show that absolute(2m-1/m+2)=1 and find all possible values of m

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Question 1140357: The acute angle between the line x-2y+3=0 and the line y=mx is 45 degrees. Show that absolute(2m-1/m+2)=1 and find all possible values of m
Answer by ikleyn(52860)   (Show Source): You can put this solution on YOUR website!
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The line x - 2y + 3 = 0  has the slope-intercept form  y =   and has, therefore, the slope  .


So, it makes the angle    with the horizontal axis "x"  such that   = .                   (1)


The line  y = mx  makes the angle    with the horizontal axis "x"  such that   = m.         (2)


The angle   -  is equal to 45 degree, according to the condition.


Hence,   = 1.


By applying the formula for tangent of the difference of arguments, you will get then


    1 =  =  = substitute here (1) and (2) to get =  = .


QED.


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