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Question
Find the joint equation of the pair of a line through the origin and perpendicular to the lines given by
2x2 - 3xy - 9y2 = 0
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Solution
Comparing the equation 2x2 - 3xy - 9y2 = 0 with ax2 + 2hxy + by2 = 0, we get,
a = 2, 2h = - 3, b = - 9
Let m1 and m2 be the slopes of the lines represented by 2x2 - 3xy - 9y2 = 0
∴ m1 + m2 = `(-"2h")/"b" = -3/9` and m1m2 = `"a"/"b" = -2/9` ...(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 + (-3/9) "xy" + (-2/9)"y"^2 = 0` ....[By(1)]
∴ `9"x"^2 - 3"xy" - 2"y"^2 = 0`
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