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

(a) Show that the B’ezier curve given by the parametric equations in (1) has (x₁, y₁) as initial point and

(x₃,y₃) as terminal point. (b) Show that x(1/2) lies on the line segment joining (x₂, y₂) to the midpoint of the line segment joining (x₁, y₁) to (x₃, y₃).