If you have used a drawing program on a computer, you have probably worked with a cu...

If you have used a drawing program on a computer, you have probably worked with a curve known as a curve. ² Such a curve is defined parametrically by using several control points in the plane to shape the curve. In Exercises 7–12, we discuss various aspects of quadratic curves. These curves are defined by using three fixed control points (x₁, y₁), (x₂, y₂), and (x₃, y₃) and a nonnegative constant w. The curve defined by this information is given by x: [0, 1] → R², x(t) = (x(t), y(t)), where

In this problem, you will establish the geometric significance of the constant w appearing in the equations in (1).

(a) Calculate the distance a between x(1/2) and (x₂, y₂).

(b) Calculate the distance b between x(1/2) and the midpoint of the line segment joining (x₁,y₁) and (x₃,y₃).

(c) Show that w = b/a. By part (b) of Exercise 9, x(1/2) divides the line segment joining (x₂,y₂) to the midpoint of the line segment joining (x₁,y₁) to (x₃, y₃) into two pieces, and w represents the ratio of the lengths of the two pieces.