3.76: Line meets curve; perpendicular bisector

0606/13/O/N/24 — Question 8 · 12 marks

The straight line y=2x+1y = 2x + 1 intersects the curve 3x2+xy+y=153x^{2} + xy + y = 15 at the points AA and BB. The point CC with coordinates (2110, k)\left(\dfrac{21}{10},\ k\right) lies on the perpendicular bisector of ABAB.
(a)
Find the exact value of kk.
[8]
(b)
The point DD lies on the perpendicular bisector of ABAB such that its perpendicular distance from ABAB is twice that of the point CC from ABAB. Find the possible coordinates of DD.
[4]