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Question
If the equation ax2 + 5xy – 6y2 + 12x + 5y + c = 0 represents a pair of perpendicular straight lines, find a and c.
Solution
Comparing ax2 + 5xy – 6y2 + 12x + 5y + c = 0 with ax2 + 2hxy + by2 + 2gx + 2fy + c = 0
We get a = a, 2h = 5, (or) h = `5/2`, b = -6, 2g = 12 (or) g = 6, 2f = 5 (or) f = 52, c = c
Condition for pair of straight lines to be perpendicular is a + b = 0
a + (-6) = 0
a = 6
Next to find c. Condition for the given equation to represent a pair of straight lines is
`|(a,h,g),(h,b,f),(g,f,c)|` = 0
`|(6,5/2,6),(5/2,-6,5/2),(6,5/2,c)|`= 0
`|(0,0,6-c),(5/2,-6,5/2),(6,5/2,c)|` = 0
R1 → R1 – R3
Expanding along first row we get 0 – 0 + (6 – c) `[25/4 + 36] = 0`
(6-c) `[25/4 + 36]` = 0
6 – c = 0
6 = c (or) c = 6
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