Answer
(a) our problem can be stated:
minimize $f\left( {x,y} \right) = \sqrt {{{\left( {x - {x_A}} \right)}^2} + {{\left( {y - {y_A}} \right)}^2}} + \sqrt {{{\left( {{x_B} - x} \right)}^2} + {{\left( {{y_B} - y} \right)}^2}} $ subject to $g\left( {x,y} \right) = 0$.
(b) using the definition of the ellipse, we explain why the level curves of $f\left( {x,y} \right)$ are ellipses.
(c) Using the method of Lagrange multipliers, we show that the ellipse through the point $P$ minimizing the length of the path is tangent to the river.
(d) Using the result in part (c), we identify the point on the river in Figure 18.
Work Step by Step
(a) Define a function
$f\left( {x,y} \right) = AP + PB$, ${\ \ \ }$ where $P = \left( {x,y} \right)$
Let the coordinates of $A$ and $B$ be $\left( {{x_A},{y_A}} \right)$ and $\left( {{x_B},{y_B}} \right)$, respectively. Thus,
$f\left( {x,y} \right) = \sqrt {{{\left( {x - {x_A}} \right)}^2} + {{\left( {y - {y_A}} \right)}^2}} + \sqrt {{{\left( {{x_B} - x} \right)}^2} + {{\left( {{y_B} - y} \right)}^2}} $
Assuming that the river is given by an equation $g\left( {x,y} \right) = 0$, our problem can be stated:
minimize $f\left( {x,y} \right) = \sqrt {{{\left( {x - {x_A}} \right)}^2} + {{\left( {y - {y_A}} \right)}^2}} + \sqrt {{{\left( {{x_B} - x} \right)}^2} + {{\left( {{y_B} - y} \right)}^2}} $ subject to $g\left( {x,y} \right) = 0$.
(b) Recall the definition of an ellipse in Section 12.5 (on page 626):
"An ellipse is an oval-shaped curve consisting of all points $P$ such that the sum of the distances to two fixed points $A$ and $B$ is a constant $K>0$".
If we write
$f\left( {x,y} \right) = \sqrt {{{\left( {x - {x_A}} \right)}^2} + {{\left( {y - {y_A}} \right)}^2}} + \sqrt {{{\left( {{x_B} - x} \right)}^2} + {{\left( {{y_B} - y} \right)}^2}} = K$,
where $K>0$, then by definition, the level curves of $f\left( {x,y} \right)$ are ellipses.
(c) By assumption, the river is given by an equation $g\left( {x,y} \right) = 0$. So, $\nabla g$ is perpendicular to the tangent to the river.
Using the method of Lagrange multipliers, we have $\nabla f = \lambda \nabla g$.
This implies that $\nabla f$ is also perpendicular to the tangent to the river. Since the level curves of $f\left( {x,y} \right)$ are ellipses, we conclude that the ellipse through the point $P$ minimizing the length of the path is tangent to the river.
(d) Using the result in part (c), the length is minimal if the ellipse through the point $P$ is tangent to the river. In such way we identify the point on the river in Figure 18.
