given point (2,6) and a line passes through point (3,0)

Question

The question is: does the distance between the point $(2,6)$ to the that line could be $5$?

is there a solution to the problem without computing?

i would glad to know.

thanks.

Answer 1

since the line $y=0$ has distance 6 from the point $(2,6)$ and there is a line with distance 0 (the one through the two points) and the distance is a continuous function, yes: there are (two) lines with distance 5. If you want to know which lines, that's another matter.

Answer 2

Draw a circle with center $(2,6)$ and radius $5$

We need to show the existence of a tangent passing through $(3,0)$

As the distance between the points $(3,0),(2,6)$ is $\sqrt{(3-2)^2+(0-6)^2}>5,$ the point $(3,0)$ is outside the circle, we know there are two tangents

Answer 3

Yes, because the difference in $y$-coordinates is more than 5, so the distance is also more than 5 (I hope that's not too much computing!).

If the points would be $(3,0)$ and $(7,4)$, this simple argument doesn't work anymore and you need to start using Pythagoras.