The eliminant of θ from x cosθ - y sin θ = 2 and x sin θ + y cos θ = 4 will give

Given that :

x cosθ - y sin θ = 2 ..............(1)

x sin θ + y cos θ = 4 .............(2)

Squaring both the equations and adding them.

(x cosθ - y sin θ)$$^{2}$$ + (x sin θ + y cos θ)$$^{2}$$ = 4+16

x$$^{2}$$ cos$$^{2}$$ θ - 2xy cosθ sinθ + y$$^{2}$$ sin$$^{2}$$ θ + y$$^{2}$$ cos$$^{2}$$ θ + 2xy cosθ sinθ + x$$^{2}$$ sin$$^{2}$$ θ = 20

x$$^{2}$$ cos$$^{2}$$ θ + y$$^{2}$$ sin$$^{2}$$ θ + y$$^{2}$$ cos$$^{2}$$ θ + x$$^{2}$$ sin$$^{2}$$ θ = 20

x$$^{2}$$ cos$$^{2}$$ θ + x$$^{2}$$ sin$$^{2}$$ θ + y$$^{2}$$ sin$$^{2}$$ θ + y$$^{2}$$ cos$$^{2}$$ θ = 20

 $$x^{2} + y^{2} = 20$$