Let f(x) = 3x2+ 8x + (2k + 1) and α and β be its zeroes
Here a = 3, b = 8 and c = 2k + 1
Given,
α = 7β -(i)
Sum of roots, α + β = -b/a
7β + β = -8/3 [Using (i)]
8β = -8/3
β = -1/3
Putting β = -1/3 in (i), we have
α = 7*-1/3 = -7/3
So, the zeroes are α = -7/3 and β = -1/3
Now,
Product of roots = -1/3*-7/3
c/a = 7/9
(2k + 1)/3 = 7/9
2k + 1 = 7/3
2k = 4/3
k = 2/3
So, value of k = 2/3