Solve the following problem step by step

Question

Determine the coordinates of points A and B on a parabola y² = x such that the point (2, 1) is on the segment AB and the midpoint of AB is as close as possible to the ordinate axis.

Answer 1

The algebra is a nuisance, but the method is straight forward.

The line through (1,2) is m=(sqrt(x)-1)/(x-2)

If you solve that for points A and B (i.e. the intersection with the parabola), you get

y_A = (1+a)/2m and y_B=(1-a)/2m where a is the discriminant of a quadratic equation such that

a² = 1-4m(-2m+1)

and y_B=(1-a)/2m

The abscissa of the midpoint is .5(1+a²)/4m²

If you differentiate this midpoint with respect to m, you will find a minimum for the abscissa when m=1 and with that slope the distance of the midpoint to the y-axis is 1.5.