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Question
Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by the following equation:
3x2 − 4xy = 0
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Solution
Consider 3x2 − 4xy = 0
∴ x(3x − 4y) = 0
∴ separate equations of the lines are x = 0 and 3x − 4y = 0
Let m1 and m2 be the slopes of these lines.
Then m1 does not exist and m2 = `3/4`
Now, required lines are perpendicular to these lines.
∴ their slopes are `- 1/"m"_1` and `- 1/"m"_2`
Since m1 does not exist, `- 1/"m"_1 = 0`
Also, m2 = `3/4, - 1/"m"_2 = - 4/3`
Since these lines are passing through the origin, their
separate equations are y = 0 and y = `- 4/3`x, i.e. 4x + 3y = 0
∴ their combined equation is
y(4x + 3y) = 0
∴ 4xy + 3y2 = 0
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