Your initial formula was correct, however, you made a few mistakes in the working.

(you forgot to multiply 4 to x²) =

(you added the 4y, instead of subtracting)

Divide everything by 3 to get the answer required

Part B is relatively straight forward. Substitute point Q into the equation of the locus and check if equality holds (it should)

Then simply find the gradient between two points you already know (B and Q) and then use the point gradient formula.