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प्रश्न
Show that the equation 2x2 - xy - 3y2 - 6x + 19y - 20 = 0 represents a pair of lines.
उत्तर
Comparing the equation
2x2 - xy - 3y2 - 6x + 19y - 20 = 0
with ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, we get,
a = 2, h = `-1/2`, b = -3, f = `19/2` and c = - 20
∴ D = `|("a","h","g"),("h","b","f"),("g","f","c")|`
`= |(2, -1/2, -3),(-1/2, -3, 19/2),(-3, 19/2, -20)|` ...[Taking `1/2` common from each row, we get]
D = ` |(4,-1,-6),(-1,-6,19),(-6,19,-40)|`
D = `1/8 |(4,-1,-6),(-1,-6,19),(-6,19,-40)|` ...`["Multiplying by" 1/8]`
`= 1/8 [4(240 - 361) + 1(40 + 114) - 6(-19 -36)]`
`= 1/8 [4(-121) + 154 - 6(- 55)]`
`= 11/8[4(-11) + 14-6(-5)]`
`= 11/8(- 44 + 14 + 30) = 0`
Also, h2 - ab = `(- 1/2)^2 - 2(- 3) = 1/4 + 6 = 25/4 > 0`
∴ The given equation represents a pair of lines.
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