Question

The angle between the pair of the lines 2*x^2+5*x*y+2*y^2+3*x+3*y+1=0 is

Answer 1

Answer:

θ = Tan⁻¹ (3/4)

Step-by-step explanation:

Let the lines represented by this equation be y = m₁x + c₁ and y = m₂x + c₂.

Combined equation for two lines = (y - m₁x + c₁ )(y - m₂x + c₂) = 0

= m₁m₂x² - (m₁+m₂)xy + y² + (m₁c₂+m₂c₁)x - (c₁ + c₂)y + c₁c₂ = 0 ----- [1]

The given equation is 2x² + 5xy + 2y² + 3x + 3y + 1  = 0

= x² + (5/2)xy + y² + (3/2)x + (3/2)y + 1/2 =0  (∵ dividing by 2) ------------- [2]

Comparing equations [1] and [2]....

m₁m₂ = 1  and m₁ + m₂ = - 5/2

(m₁ - m₂)² = (m₁ + m₂)² - 4m₁m₂ = 25/4 - 4 = 9/4

=> | m₁ - m₂ | = 3/2   (Modulus of m₁ - m₂)

Acute angle between two straight lines given by

 tanθ = | m₁ - m₂| / | 1 + m₁m₂| = 3/2/2 = 3/4.

=> θ = Tan⁻¹ (3/4)

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