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Solution for The Xy-plane Divides the Line Joining the Points (−1, 3, 4) and (2, −5, 6) (A) Internally in the Ratio 2 : 3 (B) Externally in the Ratio 2 : 3 (C) Internally in the Ratio 3 : 2 - CBSE (Science) Class 12 - Mathematics

ConceptDirection Cosines and Direction Ratios of a Line

Question

The xy-plane divides the line joining the points (−1, 3, 4) and (2, −5, 6)

• internally in the ratio 2 : 3

• externally in the ratio 2 : 3

• internally in the ratio 3 : 2

• externally in the ratio 3 : 2

Solution

$\left( b \right) \text{ externally in the ratio 2: 3 }$

$\text{ Let the XY - plane divide the line segment joining points }P\left( - 1, 3, 4 \right) \text{ and } Q\left( 2, - 5, 6 \right) \text{ in the ratio k: 1 }.$

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

$\left( \frac{k\left( 2 \right) - 1}{k + 1}, \frac{k\left( - 5 \right) + 3}{k + 1}, \frac{k\left( 6 \right) + 4}{k + 1} \right)$

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

$\Rightarrow \frac{k\left( 6 \right) + 4}{k + 1} = 0$

$\Rightarrow 6k + 4 = 0$

$\Rightarrow k = - \frac{2}{3}$

$\text{ Thus, the XY - plane divides the line segment joining the given points in the ratio 2: 3 externally } .$

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Solution The Xy-plane Divides the Line Joining the Points (−1, 3, 4) and (2, −5, 6) (A) Internally in the Ratio 2 : 3 (B) Externally in the Ratio 2 : 3 (C) Internally in the Ratio 3 : 2 Concept: Direction Cosines and Direction Ratios of a Line.
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