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Question
Find p and q, if the following equation represents a pair of perpendicular lines
6x2 + 5xy – py2 + 7x + qy – 5 = 0
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Solution
The equation of the given pair of straight lines is
6x2 + 5xy – py2 + 7x + qy – 5 = 0 ......(1)
Given that equation (1) represents a pair of perpendicular straight lines.
∴ Coefficient of x2 + coefficient of y2 = 0
6 – p = 0
⇒ p = 6
6x2 + 5xy – 6y2 = 6x2 + 9xy – 4xy – 6y2
= 3x(2x + 3y) – 2y (2x + 3y)
= (2x + 3y)(3x – 2y)
Let the separate equation of the straight lines be
2x + 3y + 1 = 0 and 3x – 2y + m = 0
6x2 + 5xy – 6y2 + 7x + qy – 5
= (2x + 3y + 1)(3x – 2y + m)
Comparing the coefficients of x, y and constant terms on both sides
2m + 3l = 7 ......(2)
3m – 2l = q ......(3)
lm = – 5 ......(4)
Equation (4)
⇒ l = 1
m = – 5
or
l = – 1
m = 5
When l = 1
m = – 5 ,
Equation (2) does not satisfy.
∴ l = – 1
m = 5
Substituting in equation (3)
3(5) – 2(–1) = q
⇒ q = 17
∴ The required values are p = 6, q = 17
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