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Question
Prove that the straight lines joining the origin to the points of intersection of 3x2 + 5xy – 3y2 + 2x + 3y = 0 and 3x – 2y – 1 = 0 are at right angles.
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Solution
Homogenizing the given equations
3x2 + 5xy – 3y2 + 2x + 3y = 0 and 3x – 2y – 1 = 0
(i.e) 3x – 2y = 1
We get (3x2 + 5xy – 3y2) + (2x + 3y)(1) = 0
(i.e) (3x2 + 5xy – 3y2) + (2x + 3y)(3x – 2y) = 0
3x2 + 5xy – 3y2 + bx2 – 4xy + 9xy – 6y2 = 0
9x2 + 10xy – 9y2 = 0
Coefficient of x2 + coefficient of y2 = 9 – 9 = 0
⇒ The pair of straight lines are at right angles.
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