Question

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

Answer 1

Explanation:

Before we do anything, let convert the line to slope intercept form

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

Now let understand the methodology of this problem:

The shortest distance between a point and a line is a line segment that is perpendicular to the line that also pass through the point.

So, this is what we need to do,

1. Find a line that is perpendicular to the original line and pass through the point (0,-3)

2. Find the intersection point of the two lines.

Step 1:

The slope of the new line should be -1/2 since that is the negative reciprocal of the original line slope.

The point should pass through (0,-3) , so let use point slope form.

`y-y_1=m(x-x_1)\\y+3=-(1/2)(x)\\y=-(1)/(2) x-3\\`

Step 2: Find the intersection point of the lines.

`-(1)/(2) x-3= 2x-(2)/(3) \\-3x-18=12x-4\\-14=15x\\x=-14/15`

Find the y coordinate of the point by plugging in (-14/15) for the function in either.

`-0.5(-(14)/(15) )-3 =(14)/(30) -(90)/(30) =-(76)/(30) =-(38)/(15)`

So the point on the line, that is closest to the point (0,-3),

(-14/15, -38/15)