Question

Find the point on the line -6x + 3y + 2 = 0 which is closest to the point (0,-3)

Answer 1

(-14/15, -114/45)

Explanation:

The shortest distance between a given point and a line is the perpendicular segment from the closest point on the line to that given point. Therefore, if we get the given equation to be in slope-intercept form and then find the line perpendicular to it that also passes (0,-3), we'll be able to see what the closest point is.

Original line in slope-intercept form

`-6x+3y+2=0\\3y+2=6x\\3y=6x-2\\y=2x-(2)/(3)`

Perpendicular line that passes through (0,-3)

`y=-(1)/(2)x+b\\-3=-(1)/(2)(0)+b\\-3=b\\\\y=-(1)/(2)x-3`

Find where both lines intersect

`2x-(2)/(3)=-(1)/(2)x-3\\12x-4=-3x-18\\15x-4=-18\\15x=-14\\x=-(14)/(15)\\\\-6x+3y+2=0\\-6(-(14)/(15))+3y+2=0\\(84)/(15)+3y+2=0\\(114)/(15)=-3y\\y=-(114)/(45)`

Therefore, the point on the line -6x + 3y + 2 = 0 which is closest to the point (0,-3) would be (-14/15, -114/45).