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Find the combined equation of the pair of a line passing through the origin and perpendicular to the line represented by following equation: 5x2 + 2xy - 3y2 = 0 - Mathematics and Statistics

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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 

Sum
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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 

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Notes

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Chapter 4: Pair of Straight Lines - Exercise 4.1 [Page 119]

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