Let R be a point on a variable line passing through origin and which cuts the lines 2x+ 5y-3 =0 and 4x+7y-3=0 at P,Q such that OP,OR,OQ are in harmonic progression. Then the locus of R

let R=rcosθ,rsinθ, P=r1cosθ,r1sinθ ,Q=r2cosθ,r2sinθ                                                       parametric coordinatesP lies on the line 2x+5y-3=0 ⇒2r1cosθ+5r1sinθ-3=0 ⇒2cosθ+5sinθ=3r1 and since Q lies on the line 4x+7y-3=0⇒4cosθ+7sinθ=3r2given that r1,r,r2 are in H.P ⇒ 2r=1r1+1r2=2cos⁡θ+5sin⁡θ3+4cos⁡θ+7sin⁡θ36=6r cosθ+ 12r sinθ⇒ x+2y=1 is the locus of R