The correct option is D (310,410)
Let the midpoint be (h,k).
Equation of chord of contact is
T=S1⇒hx+ky−2=h2+k2−2⇒hx+ky−(h2+k2)=0 ⋯(1)
Consider any point (x1,y1) on 3x+4y=10
Equation of chord of contact is
T=0⇒xx1+yy1−2=0 ⋯(2)
Comparing equation (1) and (2), we get
x1h=y1k=2h2+k2⇒x1=2hh2+k2, y1=2kh2+k2
Now putting (x1,y1) in 3x+4y=10, so
6h+8k=10h2+10k2⇒x2+y2−6x10−8y10=0
Hence, the centre is (310,410)