<p><strong>Prove that the points <math><mrow><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>7</mn></mrow><mo>)</mo></mrow></mrow></math> , <math><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>9</mn></mrow><mo>)</mo></mrow></mrow></math> , and <math><mrow><mrow><mo>(</mo><mrow><mn>6</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>1</mn><mn>3</mn></mrow><mo>)</mo></mrow></mrow></math> are collinear. Find the ratio in which the first point divides the join of the other two.</strong></p>

This is a long answer type question as classified in NCERT ExemplarLet the given points are A(0,−1,−7), B(2,1,−9), C(6,5,−13) AB=(2−0)2+(1+1)2+(−9+7)2=4+4+4=12=23 BC=(6−2)2+(5−1)2+(−13+9)2=16+16+16=48=43 AC=(6−0)2+(5+1)2+(−13+7)2=36+36+36=108=63 23+43=63i.e., AB+BC=AC∴ AB:AC=23:63=1:3Hence, point A divides B and C in 1:3 externally.

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Maths Class 12th Exemplar Solution Introduction to Three Dimensional Geometry Maths NCERT Exemplar Solutions Class 12th Chapter Twelve

Prove that the points $(0, 1, 7)$ , $(2, 1, 9)$ , and $(6, 5, 13)$ are collinear. Find the ratio in which the first point divides the join of the other two.

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Payal Gupta | Contributor-Level 10

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This is a long answer type question as classified in NCERT Exemplar

$\begin{array}{l} L e t t h e g i v e n points a r e A (0, - 1, - 7), B (2, 1, - 9), C (6, 5, - 13) \\ A B = \sqrt[]{{(2 - 0)}^{2} + {(1 + 1)}^{2} + {(- 9 + 7)}^{2}} = \sqrt[]{4 + 4 + 4} = \sqrt[]{12} = 2 \sqrt[]{3} \\ B C = \sqrt[]{{(6 - 2)}^{2} + {(5 - 1)}^{2} + {(- 13 + 9)}^{2}} = \sqrt[]{16 + 16 + 16} = \sqrt[]{48} = 4 \sqrt[]{3} \\ A C = \sqrt[]{{(6 - 0)}^{2} + {(5 + 1)}^{2} + {(- 13 + 7)}^{2}} = \sqrt[]{36 + 36 + 36} = \sqrt[]{108} = 6 \sqrt[]{3} \\ 2 \sqrt[]{3} + 4 \sqrt[]{3} = 6 \sqrt[]{3} \\ i . e ., A B + B C = A C \\ ∴ A B : A C = 2 \sqrt[]{3} : 6 \sqrt[]{3} = 1 : 3 \\ H e n c e, point A d i v i d e s B a n d C i n 1 : 3 e x t e r n a l l y . \end{array}$

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Prove that the points $(0, 1, 7)$ , $(2, 1, 9)$ , and $(6, 5, 13)$ are collinear. Find the ratio in which the first point divides the join of the other two.

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Prove that the points (0,1,7) , (2,1,9) , and (6,5,13) are collinear. Find the ratio in which the first point divides the join of the other two.

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Prove that the points $(0, 1, 7)$ , $(2, 1, 9)$ , and $(6, 5, 13)$ are collinear. Find the ratio in which the first point divides the join of the other two.