The line x-2y=1 intercepts the parabola (a U-shaped curve lying on its side with the x axis as the axis of symmetry) at two points. x=y^2/k and x=2y+1, so 2y+1=y^2/k.
Multiply through by k: 2ky+k=y^2; y^2-2ky-k=0; y^2-2ky+k^2-k^2-k=0;
(y-k)^2=k^2+k; so y-k=+sqrt(k^2+k) and
y=k+sqrt(k^2+k) (x=1+2k+2sqrt(k^2+k)) and
y=k-sqrt(k^2+k) (x=1+2k-2sqrt(k^2+k))
represent the two points of intersection.
The difference between the two y values is: 2sqrt(k^2+k) and the difference between the x values is: 4sqrt(k^2+k). The square of the length of the line segment (=4^2=16) is the sum of the squares of these differences (Pythagoras), so we can write:
16=4(k^2+k)+16(k^2+k)=20(k^2+k); dividing both sides by 4 and rearranging: 5k^2+5k-4=0;
so k=(-5+sqrt(25+80))/10=-(5+sqrt(105))/10=0.5247 or -1.5247.