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प्रश्न
Show that the equation x2 + 2xy + 2y2 + 2x + 2y + 1 = 0 does not represent a pair of lines.
उत्तर
Comparing the equation
x2 + 2xy + 2y2 + 2x + 2y + 1 = 0 with
ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, we get,
a = 1, h = 1, b = 2, g = 1, f = 1, c = 1.
The given equation represents a pair of lines, if
D = `|("a","h","g"),("h","b","f"),("g","f","c")| = 0` and h2 - ab ≥ 0
Now, D = `|("a","h","g"),("h","b","f"),("g","f","c")| = |(1,1,1),(1,2,1),(1,1,1)|`
= 1(2 - 1) - 1(1 - 1) + 1(1 - 2)
= 1 - 0 - 1 = 0
and h2 - ab = (1)2 - 1(2) = - 1 < 0
∴ given equation does not represent a pair of lines.
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