Let P = 1,0 and Q = - 1,0 are two fixed points and R is a variable point on one side of the line P Q such that ∠ R P Q - ∠ R Q P = 45 o , then the locus of R is

Question

Let P = 1,0 and Q = - 1,0 are two fixed points and R is a variable point on one side of the line P Q such that ∠ R P Q - ∠ R Q P = 45 o , then the locus of R is, y 2 - x 2 + 2 x y - 1 = 0, x 2 - y 2 + 2 x y + 1 = 0, y 2 + x 2 - 2 x y = 1, y 2 - x 2 - 2 x y + 1 = 0

MARKS App · Accepted Answer

Let, ∠ R P Q = θ & ∠ R Q P = ϕ , where θ - ϕ = 4 5 ∘ Now, t a n θ - ϕ = 1 ⇒ tan ⁡ θ - tan ⁡ ϕ 1 + tan ⁡ θ tan ⁡ ϕ = 1 , where tan ⁡ θ = y 1 - x & tan ⁡ ϕ = y x + 1 ⇒ y 1 - x - y x + 1 = 1 + y 1 - x y x + 1 ⇒ 2 y x = 1 - x 2 + y 2 ⇒ y 2 - x 2 - 2 x y + 1 = 0

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