Question

From the point \( (3,-4) \) perpendicular lines \( L_{1} \) and \( L_{2} \) are drawn on each of the lines \[ S \equiv 2 x^{2}+3 x y-2 y^{2}-7 x+y+3=0 . \]. The area of the quadrilateral formed by the pair of lines \( S=0, L_{1} \) and \( L_{2} \) is (in square units)

Answer 1

S = 2x² + 3xy - 2y² - 7x + y + 3 = 0

Consider, 2x² + 3xy - 2y² = 0

⇒ (x + 2y) (2x - y) = 0

⇒ x + 2y = 0, 2x - y = 0 

Now, (x + 2y + c₁) (2x - y + c₂) ≡ S

Comparing coefficient of x & y

2c₁ + c₂ = -7, 2c₂ - c₁ = 1

⇒ c₂ = -1, c₁ = -3

Equation of lines perpendicular to S = 0 & passing through (3, -4) are

2x - y + k₁ = 0, x + 2y + k₂ = 0

passes through (3, -4)

⇒ 6 + 4 + k₁ = 0, 3 - 8 + k₂ = 0

⇒ k₁ = -10, k₂ = 5

Lines are 2x - y - 10 = 0,