What is the value of a if 3x + 2y = 3 and ax + y = 0 form an angle of 45 degrees.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The slope of the line 3x + 2y = 3 can be determined by expressing this in the slope intercept form y = mx + c

3x + 2y = 3

=> 2y = 3 - 3x

=> y = (-3/2)y + 3

The slope of the line is -3/2.

The slope of ax + y = 0 is -a.

As the lines have to intersect and form an angle of 45 degrees, we have:

`tan 45 = (-a + (3/2))/(1 + (3/2)*a)`

=> `1 + (3/2)*a = -a + (3/2)`

=> `(5/2)*a = (1/2)`

=> `a = 1/5`

Also, `tan 45 = (-3/2 + a)/(1 + (3/2)*a)`

=> `1 + (3/2)a = -3/2 + a`

=> a = -5

The values that a can take are a = -1/5 and a = -5

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