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Question
Answer the following question:
Two lines passing through M(2, 3) intersect each other at an angle of 45°. If slope of one line is 2, find the equation of the other line.
Solution
Let m be the slope of the other line passing through M(2, 3) and making an angle of 45° with the line whose slope is 2.
∴ tan 45° = `|("m" - 2)/(1 + "m"(2))|`
∴ 1 = `|("m" - 2)/(1 + 2"m")|`
∴ `("m" - 2)/(1 + 2"m")` = ± 1
∴ `("m" - 2)/(1 + 2"m") = 1 or ("m" - 2)/(1 + 2"m")` = – 1
∴ m – 2 = 1 + 2m or m – 2 = – 1 – 2m
∴ m = – 3 or 3m = 1
∴ m = – 3 or m = `1/3`
When m = – 3, equation of the line is
y – 3 = – 3(x – 2)
∴ y – 3 = – 3x + 6
∴ 3x + y = 9
When m = `1/3`, equation of the line is
y – 3 = `1/3(x - 2)`
∴ 3y – 9 = x – 2
∴ x – 3y + 7 = 0
Hence, equations of required lines are
3x + y = 9 and x – 3y + 7 = 0.
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