JEE Advanced 2022 Paper 1 · Mathematics · Three Dimensional Geometry · medium

A plane contains the line (x-1)/(2)=(y+1)/(1)=(z-2)/(-1) and the point (1,2,3). Find the volume of the tetrahedron bounded by this plane and the…

Numerical answer

A plane contains the line

\dfrac{x-1}{2}=\dfrac{y+1}{1}=\dfrac{z-2}{-1}

and the point

(1,2,3)

. Find the volume of the tetrahedron bounded by this plane and the three coordinate planes. Give the answer correct to two decimal places.

Show answer and solution

Answer: 20.25

The plane passes through

P_0(1,-1,2)

with direction

\vec d=(2,1,-1)

and through

Q(1,2,3)

. A normal is

\vec n=\vec d\times\overrightarrow{P_0Q}=(2,1,-1)\times(0,3,1)=(4,-2,6)

, so the plane is

2x-y+3z=9

(check:

P_0

gives

2+1+6=9

Q

gives

2-2+9=9

). Intercepts:

x=\frac92,\ y=-9,\ z=3

. The volume is

\frac16\left|\frac92\cdot(-9)\cdot3\right|=\frac{121.5}{6}=\frac{81}{4}=20.25

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