Let R₁ and R₂ be two relations defined on R R as follows : (x₁, y₁) R₁ (x₂, y₂) x₁^2 + y₁^2 = x₂^2 + y₂^2 (x₁, y₁) R₂ (x₂, y₂) x₁ + y₁ = x₂ + y₂ Let E₁ be the equivalence class of the point (4, 0) under the relation R₁ , and E₂ be the equivalence class of the point (a, b) under the relation R₂ . If the intersection of E₁ and E₂ contains exactly one element, then the value of (a + b)^2 is

Step-by-step solution

The equivalence class E₁ of the point (4, 0) under R₁ is given by : x^2 + y^2 = 4^2 + 0^2 x^2 + y^2 = 16 This represents a circle centered at the origin (0, 0) with radius r = 4 . The equivalence class E₂ of the point (a, b) under R₂ is given by : x + y = a + b This represents a straight line. Since the intersection of E₁ and E₂ contains exactly one element, the line x + y - (a + b) = 0 must be tangent to the circle x^2 + y^2 = 16 . For tangency, the perpendicular distance from the center of the circle to the line