The number of normal(s) to hyperbola which can touch its conjugate is equal to

Step-by-step solution

Normal to hyperbola at t is y - c t = t ^2( x - ct ) ...(1) It touches x y=-c^2 Solving with (1), we get x c t +t^2(x-c t) +c^2=0 t^2 x^2+ ( c t -c t^3 ) x+c^2=0 For equal roots, we have ( 1 t -t^3 )^2-4 t^2=0 (1-t^4 )^2-4 t^4=0 t^4-1-2 t^2=0 or t^4-1+2 t^2=0 t ^2= 2 8 2 or -2 8 2 t ^2=1+ 2 or -1+ 2