HW22 Optımization: Proolem 2
(1 point)

Find the point on the line $2 x + 1 y - 2 = 0$ which is closest to the point $(- 5, - 1)$ .
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Answer

$(\frac{1}{5}, \frac{8}{5})$ is the point on the line closest to $(- 5, - 1)$

Steps

Step 1 ：Let the point on the line $2 x + y - 2 = 0$ be $(x, y)$

Step 2 ：Substitute $y$ from the line equation to get $y = - 2 x + 2$

Step 3 ：Use the distance formula squared to get the function to minimize: $(x - (- 5))^{2} + (- 2 x + 2 - (- 1))^{2}$

Step 4 ：Simplify the function to get: $(x + 5)^{2} + (3 - 2 x)^{2}$

Step 5 ：Take the derivative of the function with respect to $x$ : $10 x - 2$

Step 6 ：Set the derivative equal to zero and solve for $x$ : $10 x - 2 = 0$ , $x = \frac{1}{5}$

Step 7 ：Substitute $x$ back into the line equation to find $y$ : $y = - 2 (\frac{1}{5}) + 2 = \frac{8}{5}$

Step 8 ： $(\frac{1}{5}, \frac{8}{5})$ is the point on the line closest to $(- 5, - 1)$