MHT-CET Full Test-8 Mathematics Que-23 Solution

Q.23. A plane is parallel to two lines whose direction ratios are 1,0,-1and -1,1,0 and it contains the point (1,1,1) .If it cuts the co-ordinate axes at A,B,C, then the volume of the tetrahedron OABC (in cubic units) is

A. \frac{9}{4}

B. \frac{9}{2}

C. 9

D. 27

Answer: B. \frac{9}{2}

Explanation :-

The equation of the plane passing through (1,1,1) is given as:

a(x-1) + b(y-1) + c(z-1) = 0 \quad \text{...(i)}

Since the plane is parallel to the lines having direction ratios (1,0,-1) and (-1,1,0), we get:

a - c = 0 \quad \Rightarrow \quad a = b = c \quad \text{...(ii)}

Substituting in equation (i), we get:

x - 1 + y - 1 + z - 1 = 0

x+y+z=3

\frac{x}{3}+\frac{y}{3}+\frac{z}{3}=1

Co-ordinates of A,B,C are (3,0,0) , (0,3,0) and (0,0,3) respectively.

Volume of tetrahedron OABC = \frac{1}{6}\begin{vmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \\ \end{vmatrix}

=\frac{1}{6}\times 27=\frac{9}{2}cu.units