Question

find the point on the line whose equation is 2x + y - 4 = 0 that is closest to the origin.
the point on the line whose equation is 2x + y - 4 = 0 that is closest to the origin is
(simplify your answer. type an ordered pair, using integers or fractions.)

Explanation:

Step1: Rewrite the line equation

Rewrite $2x + y-4 = 0$ as $y=-2x + 4$. Let the point on the line be $(x,y)=(x,-2x + 4)$.

Step2: Define the distance function

The distance $d$ from the point $(x,-2x + 4)$ to the origin $(0,0)$ is given by the distance formula $d=\sqrt{(x - 0)^2+(-2x + 4-0)^2}=\sqrt{x^{2}+(-2x + 4)^{2}}$. To simplify calculations, we can minimize the square of the distance function $f(x)=x^{2}+(-2x + 4)^{2}=x^{2}+4x^{2}-16x + 16=5x^{2}-16x + 16$.

Step3: Find the derivative

Differentiate $f(x)$ with respect to $x$. Using the power - rule, $f^\prime(x)=10x-16$.

Step4: Set the derivative equal to zero

Set $f^\prime(x)=0$, so $10x-16 = 0$. Solving for $x$ gives $10x=16$, and $x=\frac{8}{5}$.

Step5: Find the y - coordinate

Substitute $x = \frac{8}{5}$ into $y=-2x + 4$. Then $y=-2\times\frac{8}{5}+4=\frac{-16 + 20}{5}=\frac{4}{5}$.

Answer:

$(\frac{8}{5},\frac{4}{5})$