Question

Show that equation 2x² – xy – 3y² – 6x + 19y – 20 = 0 represents a pair of lines.

Answer 1

Comparing the equation

2x² – xy – 3y² – 6x + 19y – 20 = 0

with ax² + 2hxy + by² + 2gx + 2fy + c = 0, we get,

a = 2, h = \(\cfrac{-1}{2}\), b = -3, g = -3, f = \(\cfrac{19}{2}\) , c = -20.

Taking 1/2 common from each row, we get,

= 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)]

=  1/8 (-44 + 14 + 30) = 0

Also h² – ab = – 1/2(-3) = 1/4 + 6 = 25/4 > 0

∴ the given equation represents a pair of lines.