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Write the Ratio in Which the Line Segment Joining (A, B, C) and (−A, −C, −B) is Divided by the Xy-plane. - CBSE (Arts) Class 12 - Mathematics

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Question

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

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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APPEARS IN

 RD Sharma Solution for Mathematics for Class 12 by R D Sharma (Set of 2 Volume) (2018-19 Session) (2018 to Current)
Chapter 27: Direction Cosines and Direction Ratios
Ex.Very Short Answers | Q: 10 | Page no. 25
Solution Write the Ratio in Which the Line Segment Joining (A, B, C) and (−A, −C, −B) is Divided by the Xy-plane. Concept: Direction Cosines and Direction Ratios of a Line.
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