F4 Nature of roots

Let A,B = (m,0) and (n,0) respectively,
a.
m+n = 2(k+1)/k = 2 + 2/k
Given (m + n)/2 = 5,
10 = 2 + 2/k => k = 1/4
b.
y = x^2 /4 - 2(1/4 + 1)x + 4
0 = x^2 /4 - 10x/4 + 4
x^2 - 10x + 16 = 0
x = 8 or 2
Length of AB
= 8 - 2
= 6 units

Question a
y=kx^2-2(k+1)x+4 cuts the x-axis at two points A and B
i.e. kx^2-2(k+1)x+4=0 has two distinct roots.
delta > 0
4(k+1)^2 -4*4k >0
4(k^2+2k+1)- 16k>0
4k^2+8k+4 - 16K > 0
4K^2 - 8k +4 >0
k^2 -2k +1 >0
(k-1)^2 > 0
k can be any value except 1

sum of the root = 2(k+1)/k = x1+x2 =2[ (x1+x2)/2] = 2*5 =10
2k+2 =10k
8k =2
k=1/4

Question b
put k =1/4 into the equation,
1/4 x^2 - 2*(1/4 + 1) x +4 =0
x^2 - 8 *(5/4) x+ 16 =0
x^2 -10x+16=0
(x-8)(x-2)=0
A and B are (8,0) and (2,0)
the length of AB=6