What is the equation of the line forming an angle of 45 degrees with the line x + 8y = 9?  

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The line x + 8y = 9 can be rewritten in the form y = -x / 8 + 9/8 where -1/8 is the slope. The angle that is formed by the line with the positive x – axis is arc tan (-1/8).

We need to find the equation of the line which forms an angle of 45 degrees with the given line. There is no point of intersection mentioned so we can take any point that lies on the line. We take the point (9 , 0). Now the slope of this line is tan( 45 + arc tan (-1/8)) or tan( 45 – arc tan(-1/8))

tan( 45 + arc tan (-1/8)) = 7/9

tan( 45 – arc tan(-1/8)) = 9/7

The equation of the line passing through (9, 0) with the slope 7/9  is y = (7/9)(x – 9)

=> 9y = 7x – 63

=> 7x – 9y – 63 = 0

And the equation of the line passing through (9, 0) and with the slope 9/7 is

y = (9/7)(x – 9)

=> 7y = 9x – 81

=> 9x – 7y – 81 = 0

The required lines are  7x – 9y – 63 = 0 and 9x – 7y – 81 = 0

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