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प्रश्न
Find p and q, if the equation 2x2 + 8xy + py2 + qx + 2y - 15 = 0 represents a pair of parallel lines.
उत्तर
The given equation is 2x2 + 8xy + py2 + qx + 2y - 15 = 0
Comparing it with ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, we get,
a = 2, h = 4, b = p, g = `"q"/2`, f = 1, c = - 15
Since the lines are parallel, h2 = ab
∴ (4)2 = 2p
∴ p = 8
Since the given equation represents a pair of lines
D = `|("a","h","g"),("h","b","f"),("g","f","c")| = 0` where b = p = 8
i.e. `|(2,4,"q"/2),(4,8,1),("q"/2,1,-15)| = 0`
i.e. `2(- 120 - 1) - 4 (- 60 - "q"/2) + "q"/2(4 - "4q") = 0`
i.e. - 242 + 240 + 2q + 2q - 2q2 = 0
i.e. - 2q2 + 4q - 2 = 0
i.e. q2 - 2q + 1 = 0
i.e. (q - 1)2 = 0
∴ q - 1 = 0
∴ q = 1
Hence, p = 8 and q = 1
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