Solve the problem.
-Find the point on the line x - 3y = 6 that is closest to the origin.
A) $\left( - \frac { 3 } { 5 } , - \frac { 11 } { 5 } \right)$
B) $\left( \frac { 9 } { 5 } , - \frac { 7 } { 5 } \right)$
C) $\left( - \frac { 9 } { 5 } , - \frac { 13 } { 5 } \right)$
D) $\left( \frac { 3 } { 5 } , - \frac { 9 } { 5 } \right)$

Question

Solve the problem.
-Find the point on the line x - 3y = 6 that is closest to the origin. 
A) $\left( - \frac { 3 } { 5 } , - \frac { 11 } { 5 } \right)$
B) $\left( \frac { 9 } { 5 } , - \frac { 7 } { 5 } \right)$
C) $\left( - \frac { 9 } { 5 } , - \frac { 13 } { 5 } \right)$
D) $\left( \frac { 3 } { 5 } , - \frac { 9 } { 5 } \right)$

Quizplus · Accepted Answer

We want to minimize the distance between the origin and a point on the line, which is equivalent to minimizing the square of the distance. Let the point on the line be $(x,y)$, then we want to minimize $x^2+y^2$. We can use the equation of the line to eliminate $x$: $x-3y=6$ implies $x=3y+6$. Substituting into our expression to be minimized, we obtain $f(y)=10y^2+36y+36$. Completing the square (or using calculus) shows that the minimum occurs at $y=-\frac{18}{5}$, which gives $x=3y+6=-\frac{9}{5}$. Therefore, the closest point to the origin is $\boxed{	ext{(D) } \left(\frac{3}{5},-\frac{9}{5}ight)}.$

Solve the Problem (−35,−115)\left( - \frac { 3 } { 5 } , - \frac { 11 } { 5 } \right)(−53​,−511​)

Solve the Problem $\left( - \frac { 3 } { 5 } , - \frac { 11 } { 5 } \right)$