distance from point (3,-5) to the line y=6x-2

Question

Need the answer broken down.

Answer 1

The shortest distance between a point and a line is the perpendicular distance between them.

y=6x-2 has a slope (gradient) of 6, so a line perpendicular to it has a slope of -⅙. This line has to pass through P(3,-5) so we can write:

y+5=-⅙(x-3), 6y+30=-x+3, x+6y=-27 as the equation of the perpendicular.

It meets the given line at the point given by solving the simultaneous equations:

(1) 6x-y=2

(2) x+6y=-27

Multiply (1) by 6: 36x-6y=12 then add to (2):

37x=-15, x=-15/37. y=6x-2=-90/37-2=-(90+74)/37=-164/37. So the intersection point is Q(-15/37,-164/37).

We need to find the length PQ. We can use Pythagoras:

PQ=√((3+15/37)²+(-5+164/37)²)=√(11.5968+0.3221)=√11.9189=3.4524 approx.

So the shortest distance between the given line and point P is 3.4524 approx.