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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:
5x2 + 2xy - 3y2 = 0
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Solution
Comparing the equation 5x2 + 2xy - 3y2 = 0 with ax2 + 2hxy + by2 = 0, we get,
a = 5, 2h = 2, b= - 3
Let m1 and m2 be the slopes of the lines represented by 5x2 + 2xy - 3y2 = 0
∴ `"m"_1 + "m"_2 = (-2"h")/"b" = (-2)/-3 = 2/3` and `"m"_1 "m"_2 = "a"/"b" = 5/-3` ....(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 + 2/3 "xy" - 5/3 "y"^2 = 0` ...[By (1)]
∴ 3x2 + 2xy - 5y2 = 0
Notes
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