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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:
xy + y2 = 0
Solution
Comparing the equation xy + y2 = 0 with ax2 + 2hxy + by2 = 0, we get,
a = 0, 2h = 1, b= 1
Let m1 and m2 be the slopes of the lines represented by xy + y2 = 0
∴ `"m"_1 + "m"_2 = (-2"h")/"b" = (-1)/1 = -1` and `"m"_1 "m"_2 = "a"/"b" = 0/1 = 0` ....(1)
Now required lines are perpendicular to these lines
∴ their slopes are `(-1)/"m"_1` and `(-1)/"m"_2`
Since these lines are passing through the origin, their separate equations are
y = `(-1)/"m"_1 "x"` and y = `(-1)/"m"_2 "x"`
i.e. m1y = - x and m2y = - x
i.e. x + m1y = 0 and x + m2y = 0
∴ their combined equation is
(x + m1y)(x + m2y) = 0
∴ x2 + (m1 + m2)xy + m1m2y2 = 0
∴ `"x"^2 - "xy" + 0."y"^2 = 0` ...[By (1)]
∴ x2 - xy = 0
[Note: : Answer in the textbook is incorrect.]
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