Question

Find the point on the line that is closest to the point $\left(3,1\right).$

[Hint: Express the distance $d$from the point to the line as a function of $x,$and then find the minimum value of $\left[d{\left(x\right)}^2\right].$

Short answer

The point the line closest to $\left(3,1\right)$is $\left(2,2\right).$

Step-by-step solution

Step 1. Given information

We need to determine the point on the line $y=x$that is closest to the point $\left(3,1\right).$We need to express the distance $d$from the point to the line as a function of $x$and then find the minimum value of $\left[d{\left(x\right)}^2\right].$

Step 2. Simplify

We are searching for the line $y=mx+c$perpendicular to the line $y=x.$The orthogonal lines have an inversely proportional slope.

$m=-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$1$}\right..$

$m=-1.$

So, the line having the slope $1$and passing through the point $\left(3,1\right).$

$\left(y-1\right)=-1\left(x-3\right).$

$y=-x+4.$

Now to get the coordinate of the point $\left(x,x\right)$is the line $y=x,$solve.

$x=-x+4.$

$2x=4.$

$x=2.$

The point on the line closest to the$\left(3,1\right)$is$\left(2,2\right).$