Question

Show the equation 2x² + xy – y² + x + 4y – 3 = 0 represents a pair of lines. Also find the acute angle between them.

Answer 1

Comparing the equation

2x² + xy - y² + x + 4y - 3 = 0 with

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

a = 2, h = 1/2 , b = -1, g = 1/2 , f = 2, c = – 3.

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

= 1/8 [4(12 —16) — 1( —6 — 4) + 1(4 + 2)]

= 1/8 [4( – 4) – 1(-10) + 1(6)]

= 1/8 (—16 + 10 + 6) = 0

Also, h² – ab = \(\left(\cfrac{1}{2}\right)^2\) – 2 (-1) = \(\cfrac{1}{4}\) + 2 = \(\cfrac{9}{4}\) > 0

∴ the given equation represents a pair of lines. Let θ be the acute angle between the lines