the value of 'a' for which (1-2a)x²-6ax-1=0 and ax²-x+1=o have 1 common root is

the parabolas represented by the two equations intersect at one point on the x-axis

let f(x)=(1-2a)x²-6ax-1 and g(x)=ax²-x+1

points of intersection are roots of the quadratic equation f(x)=g(x).

because there is only one point of intersection the quadratic equation f(x)=g(x) has only one root. This imposes the condition that the discriminant is zero. This condition yields an equation in a. Solve this equation to find a.