If a line A B of length r moves so that A and B always lie respectively on X -axis and y=6 x , then the locus of mid-point of A B is

Step-by-step solution

Given, A lies on X -axis and B lies on y=6 x and A B=r . C is the mid-point of A B . Let A (a, 0), B (c, 6 c) and C (h, k) . Now, h= a+c 2 and 0+6 c 2 =k array ll & c= ( k 3 ) 2 h=a+ k 3 & a= (2 h- k 3 ) & A B=r array aligned & (A B)^2=r^2 (a-c)^2+(6 c)^2=r^2 & (2 h- k 3 - k 3 )^2+ ( 6 k 3 ) ^2=r^2 & (2 h- 2 k 3 )^2+(2 k)^2=r^2 & (h- k 3 )^2+k^2= r^2 4 aligned Locus of (h, k) is given by (x- y 3 )^2+y^2= r^2 4