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Question
Find the combined equation of lines passing through the origin each of which making an angle of 30° with the line 3x + 2y - 11 = 0
Solution
The slope of the line 3x + 2y - 11 = 0 is m1 = `- 3/2`
Let m be the slope of one of the lines making an angle of 30° with the line 3x + 2y - 11 = 0
The angle between the lines having slopes m and m1 is 30°.
∴ tan 30° = `|("m" - "m"_1)/(1 + "m"."m"_1)|`, where tan 30° = `1/sqrt3`
∴ `1/sqrt3 = |("m" - (- 3/2))/(1 + "m"(- 3/2))|`
∴ `1/sqrt3 = |("2m + 3")/(2 - 3"m")|`
On squaring both sides, we get,
`1/3 = ("2m + 3")^2/(2 - "3m")^2`
∴ (2 - 3m)2 = 3(2m + 3)2
∴ 4 - 12m + 9m2 = 3(4m2 + 12m + 9)
∴ 4 - 12m + 9m2 = 12m2 + 36m + 27
∴ 3m2 + 48m + 23 = 0
This is the auxiliary equation of the two lines and their joint equation is obtained by putting m = `"y"/"x"`
∴ the combined equation of the two lines is
`3("y"/"x")^2 + 48("y"/"x") + 23 = 0`
∴ `"3y"^2/"x"^2 + "48y"/"x" + 23 = 0`
∴ 3y2 + 48xy + 23x2 = 0
∴ 23x2 + 48xy + 3y2 = 0
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