The correct options are
B (3,4) D (9,12)The equation of the circle is
x2+y2−12x−16y+75=0.
Join its centre C with O.
Let OC meets the circle in A and B.
Centre of the circle is C≡(6,8) and radius AC=√36+64−75=5.
Also, OC=√36+64=10
∴OA=OC−AC=5.
Now, the point on the circle nearest to the origin is A and farthest from it is B.
∵OA:AC=5:5=1:1
⇒A is the mid-point of OC.
∴ the coordinates of A are (0+62,0+82) i.e., (3,4)
Let the coordinate of B be (h,k).
Since C is the mid point of AB
∴3+h2=6,4+k2=8⇒h=9,k=12.
∴ the coordinates are of B are (9,12).