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Ratio in Which the Xy-plane Divides the Join of (1, 2, 3) and (4, 2, 1) is (A) 3 : 1 Internally (B) 3 : 1 Externally (C) 1 : 2 Internally (D) 2 : 1 Externally - CBSE (Arts) Class 12 - Mathematics

ConceptDirection Cosines and Direction Ratios of a Line

Question

Ratio in which the xy-plane divides the join of (1, 2, 3) and (4, 2, 1) is

•  3 : 1 internally

• 3 : 1 externally

•  1 : 2 internally

• 2 : 1 externally

Solution

 3: 1 \text{ externally }

$\text{ Suppose the XY - plane divides the line segment joining the points P } \left( 1, 2, 3 \right) \text{ and Q } \left( 4, 2, 1 \right) \text{ in the ratio k: 1 } .$

$\text{ Using the section formula, the coordinates of the point of intersection are given by }$

$\left( \frac{k\left( 4 \right) + 1}{k + 1}, \frac{k\left( 2 \right) + 2}{k + 1}, \frac{k\left( 1 \right) + 3}{k + 1} \right)$

$\text{ The Z - coordinate of any point on the XY - plane is zero }.$

$\Rightarrow \frac{k\left( 1 \right) + 3}{k + 1} = 0$

$\Rightarrow k + 3 = 0$

$\Rightarrow k = - 3 = - \frac{3}{1}$

$\text{ Thus, the XY - plane divides the line segment joining the given points in the ratio 3: 1 externally } .$

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Solution Ratio in Which the Xy-plane Divides the Join of (1, 2, 3) and (4, 2, 1) is (A) 3 : 1 Internally (B) 3 : 1 Externally (C) 1 : 2 Internally (D) 2 : 1 Externally Concept: Direction Cosines and Direction Ratios of a Line.
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