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Question
Find the combined equation of the following pair of lines:
Passing through (2, 3) and perpendicular to the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0.
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Solution
Let L1 and L2 be the lines passing through the point (2, 3) and perpendicular to the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0 respectively.
Slopes of the lines 3x + 2y – 1 = 0 and x – 3y + 2 = 0 are `(-3)/2` and `(-1)/-3 = 1/3` respectively.
∴ Slopes of the lines L1 and L2 are `2/3` and –3 respectively.
Since the lines L1 and L2 pass through the point (2, 3), their equations are
y – 3 = `2/3("x" - 2)` and y – 3 = –3(x – 2)
∴ 3y – 9 = 2x – 4 and y – 3 = –3x + 6
∴ 2x – 3y + 5 = 0 and 3x + y – 9 = 0
∴ Their combined equation is (2x – 3y + 5)(3x + y – 9) = 0
∴ 6x2 + 2xy – 18x – 9xy – 3y2 + 27y + 15x + 5y – 45 = 0
∴ 6x2 – 7xy – 3y2 – 3x + 32y – 45 = 0
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