If a variable straight line passing through the point of intersection of the lines x-2 y+3=0 and 2 x-y-1=0 intersects the X , Y -axes at A and B respectively, then the equation of the locus of a point which divides the segment AB in the ratio -2: 3 is

Step-by-step solution

Since, equation of a line passing through the two given lines is x-2 y+3+K(2 x-y-1)=0 (2 K+1) x+(-K-2) y-K+3=0 (2 K+1) (K-3) x+ (-K-2) (K-3) y=1 So, A ( K-3 2 K+1 , 0 ), B (0, 3-K K+2 ) Now, (x, y)= ( 0+ 3(K-3) 2 K+1 1 , 2(K-3) K+2 1 ) aligned x & = 3 K-9 2 K+1 K= x+9 3-2 x y & = 2(K-3) K+2 K= 6+2 y 2-y aligned So, x+9 3-2 x = 6+2 y 2-y 14 x+3 x y-15 y=0 .