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प्रश्न
Find the joint equation of the pair of the line through the point (2, -1) and parallel to the lines represented by 2x2 + 3xy - 9y2 = 0.
उत्तर
The combined equation of the given lines is
2x2 + 3xy - 9y2 = 0
i.e. 2x2 + 6xy - 3xy - 9y2 = 0
i.e. 2x(x + 3y) - 3y(x + 3y) = 0
i.e. (x + 3y)(2x - 3y) = 0
∴ their separate equations are
x + 3y = 0 and 2x - 3y = 0
∴ their slopes are m1 = `(-1)/3` and m2 = `(-2)/-3 = 2/3`
The slopes of the lines parallel to these lines are m1 and m2 i.e. `- 1/3` and `2/3`.
∴ the equations of the lines with these slopes and through the point (2, -1) are
y + 1 = `- 1/3`(x - 2) and y + 1 = `2/3` (x - 2)
i.e. 3y + 3 = - x + 2 and 3y + 3 = 2x - 4
i.e. x + 3y + 1 = 0 and 2x - 3y - 7 = 0
∴ the joint equation of these lines is
(x + 3y + 1)(2x - 3y - 7) = 0
∴ 2x2 - 3xy - 7x + 6xy - 9y2 - 21y + 2x - 3y - 7 = 0
∴ 2x2 + 3xy - 9y2 - 5x - 24y - 7 = 0
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