Question

If θ is the acute angle between the lines given by 3x² – 4xy + by² = 0 and tan θ = `1/2`, find b.

Answer 1

3x² – 4xy + by² = 0

Here a = 3,

h = –2,

tan θ = `1/2`  ...(Given)

We have : tan θ = `|(2sqrt(h^2 - ab))/(a + b)|`

∴ `1/2 = |(2sqrt((-2)^2 - (3)(b)))/(3 + b)|`

Squaring both sides,

`1/4 = (4(4 - 3b))/(3 + b)^2`

∴ 9 + 6b + b² = 16(4 – 3b)

∴ 9 + 6b + b² = 64 – 48b

∴ b² + 54b – 55 = 0

∴ b² + 55b – b – 55 = 0

∴ b(b + 55) – 1(b + 55) = 0

∴ (b – 1)(b + 55) = 0

∴ b – 1 = 0 or b + 55 = 0

∴ b = 1 or b = –55