find the point on the line $y = 7x + 10$ closest to the point $(5, - 5)$. the function giving the distance…

find the point on the line $y = 7x + 10$ closest to the point $(5, - 5)$. the function giving the distance between the point and the line is $s=\text{ (enter a function of }x\text{)}$

find the point on the line $y = 7x + 10$ closest to the point $(5, - 5)$. the function giving the distance between the point and the line is $s=\text{ (enter a function of }x\text{)}$

Answer

Explanation:

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, one point is $(x,y)$ on the line $y = 7x+10$ and the other point is $(5,-5)$. So $x_1 = x$, $y_1=7x + 10$, $x_2=5$, $y_2=-5$.

Step2: Substitute into distance formula

[ \begin{align*} s&=\sqrt{(x - 5)^2+((7x + 10)-(-5))^2}\ &=\sqrt{(x - 5)^2+(7x+15)^2}\ &=\sqrt{x^{2}-10x + 25+49x^{2}+210x + 225}\ &=\sqrt{50x^{2}+200x + 250} \end{align*} ]

Answer:

$\sqrt{50x^{2}+200x + 250}$