Answer
a. $ y_{p}=-x+1$
b. $ 2(x-x^{2})$
Work Step by Step
a. First, we find the equation for the line AB. The geometry shows the coordinates of points A(1,0) and B(0,1), and the equation of AB can be found as $ y=-x+1$. As the point $ P(x_{p},y_{p})$ is on the line AB, its coordinates satisfy the same equation, so we have $ y_{p}=-x_{p}+1=-x+1$ (where $ x=x_{p}$).
b. The area of the rectangle is $ A=2y_{p}\times x_{p}=(-x_{p}+1)\times x_{p}=2(x-x^{2})$
