The circles x^2+y^2-4 x-81=0, x^2+y^2+24 x-81=0 intersect each other at points A and B . A line through point A meet one circle at P and a parallel line through B meet the other circle at Q . Then the locus of the mid point of P Q is

Step-by-step solution

Now let P (x₁, y₁ ) and Q (x₂, y₂ ) Now solve S₁ and PA then x₁= -18 m-24 m^2+1 and y₁= -9 m^2-24 m+9 m^2+1 Now solve S₂ and B Q then x₂= 18 m+4 m^2+1 and y₂= 9 m^2+4 m-9 m^2+1 Let mid pt of P Q is ( h , k ) then x ₁+ x ₂=2 ~h and y ₁+ y ₂=2 k x₁+x₂= -20 m^2+1 =2 h h k = 1 ~m y₁+y₂= -20 m m^2+1 =2 k -20 ( k h )^2+1 =2 hh (h^2+k^2+10 h )=0