Question

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

kxy + 10x + 6y + 4 = 0

Answer 1

Comparing the given equation with

ax² + 2 hxy + by² + 2gx + 2fy + c = 0,

we get, a = 0, h = k/2 , b = 0, g = 5, f = 3, c = 4

Now, given equation represents a pair of lines.

∴ abc + 2fgh – af² – bg² – ch² = 0

∴ (0)(0)(4) + 2(3)(5) (k/2) – 0(3)² – 0(5)² – 4

(k/2)² = 0

∴ 0 + 15k – 0 – 0 – k² = 0

∴ 15k – k² = 0

∴ -k(k – 15) = 0

∴ k = 0 or k = 15.

If k = 0, then the given equation becomes 10x + 6y + 4 = 0 which does not represent a pair of lines.

∴ k ≠ o

Hence, k = 15.