A stair-case of length l rests against a vertical wall and a floor of a room. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1: 2 . If the staircase begins to slide on the floor, then the locus of P is:

Step-by-step solution

Let point A (a, 0) is on x -axis and B (0, b) is on y -axis. Let P (h, k) divides AB in the ratio 1: 2 . So, by section formula aligned &h= 2(0)+1(a) 1+2 = a 3 &k= 2(b)+1(0) 3 = 2 b 3 & a=3 h and b= 3 k 2 aligned Now, a^2+b^2=l^2 aligned & 9 h^2+ 9 k^2 4 =l^2 & h^2 ( l 3 )^2 + k^2 ( 2 l 3 )^2 =1 aligned Now e = 1- ( l^2 9 9 4 l^2 ) = 1- 1 4 = 3 2 Thus, required locus of P is an ellipse with eccentricity 3 2 .