Write the ratio in which the line segment joining (a, b, c) and (-a, -c, -b) is divided by the xy-plane.

Given,

The line segment is formed by P and Q points where

Point P = (a,b,c)

Point Q = (-a,-c,-b)

From the figure, we can clearly see that, the line segment joining points P and Q is meeting the plane XY at point G.

Let Point G be (x,y,0) as the z-coordinate on xy plane does not exist.

Also let point G divides the line segment joining P and Q in the ratio m:n.

The coordinates of the point G which divides the line joining points A(x₁,y₁,z₁) and B(x₂,y₂,z₂) in the ratio m:n is given by

Here, we have m:n

x₁ = a y₁ = b z₁ = c

x₂ = -a y₂ = -c z₂ = -b

By using the above formula, we get,

Now, this is the same point as G(x,y,0),

As the x-coordinate is zero,

[Cross Multiplying]

-bm + cn = 0 × (m + n)

-bm + cn = 0

-bm = -cn

Therefore, the ratio in which the plane-XY divides the line joining P & Q is c:b