Question

If the curve x²+2y²=2 intersect the line x + y =1 at two points P and Q then the angle subtended by the line segment PQ at the origin is

Answer 1

Using homogenization

x² + 2y² = 2(1)²

⇒x² + 2y² = 2(x + y)²

⇒x² + 2y² = 2x² + 2y² + 4xy

⇒x² + 4xy = 0

for ax² + 2hxy + by² = 0, obtuse angle between lines θ is

tan θ = ±(2√(h²–ab))/(a+b)

⇒tan θ = ±4

⇒tan θ = –4

cot θ = -1/4

θ = cot^–1(–1/4)

θ = π – cot^–1 (1/4)

θ = π – (π/2– tan^–1(1/4) )

θ = π/2 + tan^-1(1/4)