Question

Find the distance from point (-1, 3) to the line 5 x - 4 y = 10.

Answer 1

We could use the formula, derive the formula, or just work it out for this case.  Let's do the latter.

The distance of a point to a line is the length of the perpendicular from the line to the point.

So we need the perpendicular to 5x-4y=10 through (-1,3).  To get the perpendicular family we swap x and y coefficients, negating one.  We get the constant straightforwardly from the point we're going through:

4x + 5y = 4(-1)+5(3) = 11

Those lines meet at the foot of the perpendicular, which is what we're after.

4x + 5y = 11

5 x - 4y = 10

We eliminate y by multiplying the first by four, the second by five and adding.

16x + 20y  = 44

25x - 20y = 50

41x = 94

x = 94/41

y  = (11 - 4x)/5 = 15/41

We want the distance from (-1,3) to (94/41,15/41)

$d = \sqrt{ (-1 - 94/41)^2 + (3 - 15/41)^2 } = \dfrac{27}{\sqrt{41}}$