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Question 339327: What is the distance of the z-intercept from the x-intercept in the equation ax+by+cz+d=0.
Answer by nyc_function(2741)   (Show Source):
You can put this solution on YOUR website! The plane crosses the x axis when y = z = 0; similarly x = y = 0 when it crosses the z-axis. We have to make some assumptions about the coefficients a and c---namely that they are not zero so that the plane does actually cross these axes. (Well, there is the other degenerate case in which the plane is the xz-plane y = 0.)
If a and c are nonzero, the intersection of the plane with the x-axis occurs at the point (-d/a,0,0) and that with the z-axis occurs at (0,0,-d/c). The distance between these points is
√((-d/a)^2 + (-d/c)^2) = |d|/√(a^2 + c^2)
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