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Question
If equation ax2 - y2 + 2y + c = 1 represents a pair of perpendicular lines, then find a and c.
Solution
The given equation represents a pair of lines perpendicular to each other.
∴ coefficient of x2 + coefficient of y2 = 0
∴ a - 1 = 0
∴ a = 1
With this value of a, the given equation is
x2 - y2 + 2y + c - 1 = 0
Comparing this equation with
Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0, we get,
A = 1, H = 0, B = -1, G = 0, F = 1, C = c - 1
Since the given equation represents a pair of lines,
D = `|("A","H","G"),("H","B","F"),("G","F","C")| = 0`
∴ `|(1,0,0),(0,-1,1),(0,1,"c - 1")| = 0`
∴ 1(- c + 1 - 1) - 0 + 0 = 0
∴ - c = 0
∴ c = 0
Hence, a = 1, c = 0.
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