SATHEE: 3D Geometry 3 Question 64

3D Geometry 3 Question 64

64. A plane is parallel to two lines whose direction ratios are $(1, 0, - 1)$ and $(- 1, 1, 0)$ and it contains the point $(1, 1, 1)$ . If it cuts coordinate axes at $A, B, C$ . Then find the volume of the tetrahedron $O A B C$ .

(2004, 2M)

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Answer:

Correct Answer: 64. (i) $x + y - 2 z = 3$ (ii) $Q (6, 5, - 2)$

Solution:

Let the equation of plane through $(1, 1, 1)$ having $a, b, c$ as DR’s of normal to plane,

$a (x - 1) + b (y - 1) + c (z - 1) = 0$

and plane is parallel to straight line having DR’s.

$(1, 0, - 1) and (- 1, 1, 0)$

$\Rightarrow$

$\begin{aligned} a - c & = 0 \\ - a + b & = 0 \\ a = b & = c \end{aligned}$

$\Rightarrow$

$∴$ Equation of plane is

$x - 1 + y - 1 + z - 1 = 0 or \frac{x}{3} + \frac{y}{3} + \frac{z}{3} = 1 .$

Its intercept on coordinate axes are

$A (3, 0, 0), B (0, 3, 0), C (0, 0, 3) .$

Hence, the volume of tetrahedron $O A B C$

$= \frac{1}{6} [a b c c] = \frac{1}{6} | \begin{array}{lll} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{array} | = \frac{27}{6} = \frac{9}{2} cu units$

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3D Geometry 3 Question 64

64. A plane is parallel to two lines whose direction ratios are $(1, 0, - 1)$ and $(- 1, 1, 0)$ and it contains the point $(1, 1, 1)$ . If it cuts coordinate axes at $A, B, C$ . Then find the volume of the tetrahedron $O A B C$ .

Answer:

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3D Geometry 3 Question 64

64. A plane is parallel to two lines whose direction ratios are (1,0,−1) and (−1,1,0) and it contains the point (1,1,1). If it cuts coordinate axes at A,B,C. Then find the volume of the tetrahedron OABC.

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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64. A plane is parallel to two lines whose direction ratios are $(1, 0, - 1)$ and $(- 1, 1, 0)$ and it contains the point $(1, 1, 1)$ . If it cuts coordinate axes at $A, B, C$ . Then find the volume of the tetrahedron $O A B C$ .