Consider the hyperbola  H:x2−y2=1and a circle S with centerNx2,0. Suppose that H and S touch each other at a point Px1,y1 with x1>1 and  y1>0.The common tangent to H and S at P intersects the x-axis at point M. If l,m is the centroid of the triangle ΔPMN, then the correct expression(s) is (are)

Tangent of the curve x2−y2=1 at Px1,y1  is xx1−yy1=1Hence, M1x1,0Equation of the circle to touhcing xx1−yy1=1 at Px1,y1 is x−x12+y−y12+2kxx1−yy1−1=0center x2,ox1−kx1=x2  and y1+ky1=0k=−1x2=2x1∴M1x1,0,N2x1,0,Px1,x12−1⇒l=133x1+1x1, m=13x12−1dldx1=1−13x12,dmdx1=x13x12−1∴dmdy1=dmdx1dy1dx1=x13x12−1x1y1=13