The angle between the lines y1 = k1*x1 and y2 = k2*x2 is 45 degrees. Find the functions if k1/k2 = 6.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The graph of the functions y1 = k1*x1 and y2 = k2*x2 are straight lines with slope k1 and k2 respectively. It is given that k1/k2 = 6 or k1 = 6*k2.

The lines are y1 = 6*k2*x1 and y2 = k2*x2.

The angle between the lines is 45 degrees.

=> 45 = `tan^-1(k2) - tan^-1(6*k2)`

=> `tan^-1(6*k2) = tan^-1(k2) - 45`

Take the tangent of both the sides

=> 6*k2 = `(tan(tan^-1 k2) - tan 45)/(1+tan(tan^-1 k2)*tan 45)`

=> 6*k2 = `(k2 - 1)/(1+k2)`

=> 6*k2 + 6*k2^2 = k2 - 1

=> 6k2^2 - 5k2 + 1 = 0

=> 6k2^2 - 3k2 - 2k2 + 1 = 0

=> 3k2*(2k2 - 1) - 1(2k2 - 1) = 0

=> (3k2 - 1)(2k2 - 1) = 0

=> k2 = 1/3 and k2 = 1/2

k1 = 2 and k1 = 3

The functions are y1 = 2*x1 and y2 = (1/3)*x2, and y1 = 3*x1 and y2 = (1/2)*x2

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