Let X~N(3,5) and Y~N(-7,2) be independent. Find values of C1,C2,C3,C4,C5,C6 such that

C1(X+C2)^C3
-------------- ~ t(C6)
(Y+C4)^C5

My attempt

so the t distribution can become X/sqrt(Y/C6)
so Y is a chi squared distribution with C6 degrees of freedom

so if I do
C2=0
C3=1
C1=sqrt(4*C6)
C4=7
C5=0.5

i get

sqrt(4*C6)(X+0)^1
-------------- ~ t(C6)
(Y+7)^0.5

X
-------------- ~ t(C6)
1/[sqrt(4*C6)] * sqrt(Y+7)

X
-------------- ~ t(C6)
sqrt([Y+7]/[4C6])

here since Y~N(-7,2), and in the equation the mean is being subtracted and then its being divided by its sd, it gets normalized

X
----------- ~ t(C6)
sqrt(Y/C6)

X
----------- = X/sqrt(Y/C6)
sqrt(Y/C6)


Is this right?
Does X have to be standardized also?