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Question
Write the ratio in which the line segment joining (a, b, c) and (−a, −c, −b) is divided by the xy-plane.
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Solution
\[ \text{ Suppose the line segment joining the points } \left( a, b, c \right) \text{ and } \left( - a, - c, - b \right) \text{ is divided by the XY - plane at a point R in the ratio } \lambda: 1 . \]
\[\text{ Coordinates of R are} \]
\[\left( \frac{\lambda\left( - a \right) + 1\left( a \right)}{\lambda + 1}, \frac{\lambda\left( - c \right) + 1\left( b \right)}{\lambda + 1}, \frac{\lambda\left( - b \right) + 1\left( c \right)}{\lambda + 1} \right)\]
\[\text{ Since R lies on XY - plane, Z - coordinate of R must be zero } . \]
\[ \Rightarrow \frac{\lambda\left( - b \right) + 1\left( c \right)}{\lambda + 1} = 0 = \frac{c}{b} \]
\[\text{ Thus, the required ratio is } \frac{c} {b: 1} \ \text{or } {c: b} . \]
\[ \text{ Hence, the XY - plane divides the line in the ratio } c: b .\]
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