Find the distance from the point (6,-2) to the line y=2x-4

Answer 1

The distance will be the shortest distance between the point (6,-2) and the line y=2x-4. That distance is a line from the point (6,-2) that is perpendicular to the line y=2x-4.

We need to first determine the equation of that perpendicular line. The slope of the line y=2x-4 is 2 (based on the slope-intercept formula y = mx + b, where m is the slope). The line perpendicular to that line will have a slope equal to the negative reciprocal, or -1/2.

We will use the point-intercept form to find the equation of the perpendicular line: (y-y1) = m (x - x1), where m = -1/2 and (x1,y1) = (6,-2)

y - 6 = -1/2 (x --2)

y - 6 = -1/2 (x + 2)

y - 6 = -1/2x - 1

y = -1/2x + 5

Now to determine where the two lines intersect set the two equations (y=2x-4 and y = -1/2x + 5) equal to each other:

2x - 4 = -1/2x + 5

5/2x = 9

x = 3.6

substituting x = 3.6 into either equation (chose y = 2x-4):

y = 2(3.6) -4 = 3.2

So our two points are (6,-2) and (3.6,3.2)

Now we just need to find the distance between those two points using the distance formula:

d = sqrt((y2-y1)^2 + (x2-x1)^2)

= sqrt( (3.2 - -2)^2 + (3.6-6)^2)

= sqrt( 5.2^2 + (-2.4)^2)

= sqrt(32.8) = 5.727

Answer 2

We can get distance easer, if we change equation of line to normal like this 2x - y - 4 = 0, then divide this equation by sqrt(2² + (- 1)²) = sqrt5 = √5. We get (2x - y - 4)/√5 = 0.

Now we substitute coordinates of this point (6, - 2):

d = abs(2·6 - (- 2) - 4)/√5 = 10/√5 = 2√5