i need help with this question. it needs to be solved by using the methods below not language interpolation.

8. In a (2, 5) Shamir secret sharing scheme with modulus 19, two of the shares are (1,11) and (2,4). Another share is (3,k), but the value of k is unreadable. Find the correct value of k. 7 marks Solution. For modulus 19, a (2, 5) Shamir secret sharing scheme uses the degree 1 polynomial s(x) = M +six (mod 19). Since (1, 11) and (2, 4) are both shares, working mod 19, M+ s = 11 M + 251 = 4. Subtracting these two equations gives $1 = -7 = 12 (mod 19) Either by using the Euclidean algorithm or noting that So the first equation above becomes M + 12 = 11 And M - 18 (mod 19) Thus, s(x) = 18 + 12x (mod 19) As (3, k) is a share, k = 18+12*3 = 18+36 =54 =16 (mod 19) So k = 16.Another example: > Assume that Nghia has set up a (3,8)-threshold scheme with modulus 37. He makes this information public. He then sends the point (1, 1) to Jo, the point (2,30) to Phil and the point (3,27) to Julia. > Jo, Phil and Julia now want to collaborate to find Nghia's secret message, which they know is the constant term in the secret quadratic equation. > They set up a general equation: s(x) = M + sx + rx2 (mod 37) where M, r and s are the constants to be determined.> They obtain the equations 1 = M + s + r (1) 30 = M + s*2 + r*4 (2) 27 = M +s*3 + r*9 (3) all modulo 37. Subtracting (1) from (2) produces 29 = s +3r (4) (mod 37); subtracting (1) from (3) produces 26 = 25 + 8r (5) (mod 37). Since gcd(2, 37) = 1, the inverse of 2 exists (and is 19) and we can multiply through equation (5) with it which then becomes 13 = s + 4r (6). Subtracting equation (4) from (6) yields 21 = r. Substituting into (4), s = 29 63 = 3. Finally, from (1), M = 1-3-21 = 14. So they has found both the secret equation 500 = 14 + 3x + 21 x2 (mod 37) and the secret message 14 = s(0). Part A: Secret Sharing 1. In a (3, 7) Shamir secret sharing scheme with modulus 59, three of the shares are (5, ll), (10, 43) and (20, 34). Another share is (7, k), but the value of k is unreadable. Find the correct value of k (3 marks). Verify your result with Maple (2 marks)