Find the point on the line y = 5x + 2 that is closest to the origin.

The shortest distance from a point to a line is the perpendicular distance. The origin is essentially a point with coordinates (0,0); hence, we have to find the equation of the line that's both perpendicular to [tex]y=5x+2[/tex] and crosses the origin.

We can do this by recognizing that slope of a perpendicular line is the opposite reciprocal of the slope of the original line.

Opposite Reciprocal of 5: [tex]-\frac{1}{5}[/tex]

Now that we have the slope and the point, we can use the point-slope form to get the equation of the perpendicular.

Point-Slope Form: [tex]y-y_1=m(x-x_1)[/tex] (m is slope, [tex]x_1[/tex] and [tex]y_1[/tex] are the point's coordinates)

[tex]y-0=-\frac{1}{5}(x-0)\\y=-\frac{1}{5}x[/tex]

Since we need to find a point on the original line that's closest to the origin, we need to find the intersection of the line and its perpendicular. We can do this by setting both lines equal to each other and solving for x and y.

[tex]5x+2=-\frac{1}{5}x\\25x+10=-x\\26x=-10\\x=-\frac{10}{26}\\x=-\frac{5}{13}[/tex]

Substituting x in any of the equations will give us y. Here, I will put it in the equation [tex]y=-\frac{1}{5}x[/tex]

[tex]y=-\frac{1}{5}*-\frac{5}{13}\\y=\frac{5}{65}\\y=\frac{1}{13}[/tex]

The closest point to the origin on the line [tex]y=5x+2[/tex] is [tex]\left(-\frac{5}{13}, \frac{1}{13}\right)[/tex]