Question

how to choose the sign of the radical in the denominator of the formula for the distance from a point to a line.

Answer 1

The distance of a points from a line is defined by

`d=\frac{\sqrt{a^(2)+b^(2) } }`

Where the line is
`ax+by+c=0` and the point is
`(x_(0) ,y_(0) )`. Also
`a \\eq 0` and
`b\\eq 0`.

Observe that the denominator has a square root. Remember that square roots can be positive or negative, because they are the opposite of a square power, which is the same no matter what sign its base has.

So, you need to consider that distance is a magnitude that is always positive, because it's an scalar magnitude, which doesn't include a sign.

Therefore, you determine the sign of the radical depending on the sign resulting in the numerator, if there's a negative sign in the numerator, then you need a negative result in the denominator (root) to make it positive, because distance is a scalar magnitude.

However, notice that the formula for the distance has the numerator as an absolute value, that means the numerator will be always positive, so you'll consider only positive results from the square root.