Question

Find the value of k if the following equations represent a pair of lines :

3x² + 10xy + 3y² + 16y + k = 0

Answer 1

Comparing the given equation with

ax² + 2hxy + by² + 2gx + 2fy + c = 0,

we get, a = 3, h = 5, b = 3, g = 0, f= 8, c = k.

Now, given equation represents a pair of lines.

∴ abc + 2fgh – af² – bg² – ch² = 0

∴ (3)(3)(k) + 2(8)(0)(5) – 3(8)² – 3(0)² – k(5)² = 0

∴ 9k + 0 – 192 – 0 – 25k = 0

∴ -16k – 192 = 0

∴ – 16k = 192

∴ k= -12.