Let L1=0, L2=0 and L3=0 are three given lines .A variable line is drawn through the origin to cut the lines at A, B, and C, respectively. If P is a point on variable line in such a way that,

7/OP=1/OA+2/OB+4/OC , then find the locus of P.

let the three lines be  y=m₁x +c₁, y=m₂x +c₂ and y=m₃x +c₃. Let variable line be y=mx. A=(c₁/(m-m₁),mc₁/(m-m₁)),

B= (c₂/(m-m₂),mc₂/(m-m₃)), and C=(c₃/(m-m₃),mc₃/(m-m₃)),. let P=(x,y)=(x,mx)

OP=√(x²(1+m²)), 1/OA= (m-m₁)/c₁(√(1+m²)), 2/OB= 2(m-m₂)/c₂(√(1+m²)) and 4/OC = 4(m-m₃)/c₃(√(1+m²))

Therefore,P is given by y(1/c₁ +2/c₂ + 4/c₃) = 7 + x(m₁/c₁ +2m₂/c₂  +4m₃/c₃) 

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