A variable line ' L ' passing through the origin cuts two parallel lines x-y+10=0 and x-y+20=0 at two points A and B respectively. If P is a point on line ' L ' such that O A, O P, O B are in harmonic progression, then the locus of P is

Step-by-step solution

Let the equation of line passing through origin is y=m x which cut the parallel lines x-y+10=0 and x-y+20=0 at points A and B respectively. Then, co-ordinates of points A and B are, gathered y=m x and x-y+10=0 x-m x=-10 x= -10 1-m = 10 m-1 and y= 10 m m-1 A ( 10 m-1 , 10 m m-1 ) gathered So, aligned O A & = 100 (m-1)^2 + 100 m^2 (m-1)^2 & = 100 (1+m^2 ) (m-1)^2 = 10 (1+m^2 ) m-1 aligned Similarly, coordinate of point B is B ( 20 m-1 , 20 m m-1 ) Then, O B= 400 (m-1)^2 + 400 m^2 (m-1)^2 = 20 1+m^2 (m-1) Let point P(