Answer:
Option A,B,D
Explanation:

Equation of family of circles touching hyperbola at (x1,y1) is
(x-x1)2+(y-y1)2+λ(x x1-y y1-1)=0
Now, its centre is (x2,0).
∴ [−(λx1−2x1)2,−(−2y1−λy1)2]=(x2,0)
⇒ 2y1+λy1=0⇒λ=−2
and 2x1−λx1=2x2⇒x2=2x1
∴ P(x1,√x21−1)
And N(x2,0)=(2x1,0)
As tangent intersect X-axis at M(1x,0)
Centroid of △PMN=(l,m)
⇒ (3x1−1x13,y1+0+03)=(l,m)
⇒l=3x1+1x13
On differentiating w.r.t x1, we get
dldx1=3−1x213
⇒ dldx1=1−13x21, for x1>1
and m=√x21−13
On differentiating w.r.t x1 , we get
dmdx1=2x12×3√x21−1=x13√x21−1, for x1>1
also m=y13
On differentiating w.r.t y1, we get
dmdy1=13, for y1>0