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Question
Prove that the combined of the pair of lines passing through the origin and perpendicular to the lines ax2 + 2hxy + by2 = 0 is bx2 - 2hxy + ay2 = 0.
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Solution
Let m1 and m2 be the slopes of the lines represented by ax2 + 2hxy + by2 = 0.
∴ m1 + m2 = `(- 2"h")/"b"` and m1m2 = `"a"/"b"` ....(1)
Now, the 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 (-"2h")/"b" "x" + "a"/"b" "y"^2 = 0` ...[By(1)]
∴ bx2 - 2hxy + ay2 = 0
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