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
\(\frac{m}{n}=\frac{c}{d}\)
Therefore, the ratio in which the plane-XY divides the line joining P & Q is c : b
