The slope of a line is 3 times the slope of another line
If the tangent of the angle between them is , 1/4 find
the slopes of the lines.

Step-by-step explanation:

Let the lesser slope be represented by x. This is also the tangent of the angle it makes with the +x axis. Then the larger slope is 3x, and likewise, its angle has the tangent 3x.

So, we can use the tangent formula for the difference of angles to find the values of these slopes.

  tan(a -b) = (tan(a) -tan(b))/(1 +tan(a)tan(b))

We want this tangent to be 1/4, so ...

  1/4 = (3x -x)/(1 +(3x)(x))

  3x^2 +1 = 4(2x) . . . . . . . cross multiply

  3x^2 -8x +1 = 0 . . . . . . . write in standard form

Now, we can use the quadratic formula to find x.

  x = (-b ±√(b^2-4ac))/(2a) = (-(-8)±√((-8)^2 -4(3)(1)))/(2(3))

  = (8 ± √52)/6

  x = (4 ±√13)/3

We're only interested in the smaller of these two values.

The slope of one line is 4 -√13; the slope of the other is (4-√13)/3.

Check

The line with the lesser slope makes an angle of arctan((4-√13)/3) ≈ 7.490° with the +x axis. The line with the greater slope makes an angle of arctan(4-√13) ≈ 21.527° with the +x axis.

The difference angle is about 14.036°, and the tangent of that is 0.25, as required.

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