Question

The foot of perpendicular from the origin $O$ to a plane $P$ which meets the co-ordinate axes at the points $A , B , C$ is $(2, a , 4), a \in N$ If the volume of the tetrahedron $OABC$ is 144 unit $^3$, then which of the following points is NOT on $P$ ?

• A. $(0,4,4)$
• B. $(3,0,4)$
• C. $(0,6,3)$
• D. $(2,2,4)$

Hint:

To determine if a point lies on a plane, substitute its coordinates into the plane equation. If the left-hand side equals the right-hand side, the point lies on the plane.

Answer 1

The Correct Option is B

Equation of Plane:
$(2\hat{i}+aj+4\hat{k})\cdot [(x-2)\hat{i}+(y-a)\hat{j}+(z-4)\hat{k}]=0$
$\Rightarrow 2x+ay+4z=20+a^2$
$\Rightarrow A\equiv \left(\frac{20+a^2}{2},0,0\right)$
$B\equiv \left(0,\frac{20+a^2}{a},0\right)$
$C\equiv \left(0,0,\frac{20+a^2}{4}\right)$
$\Rightarrow$ Volume of tetrahedron
$=\frac{1}{6}[\vec{a}\vec{b}\vec{c}]$
$=\frac{1}{6}\vec{a}\cdot (\vec{b}\times \vec{c})$
$\Rightarrow \frac{1}{6}\left(\frac{20+a^2}{2}\right)\cdot \left(\frac{20+a^2}{a}\right)\cdot \left(\frac{20+a^2}{4}\right)=144$
$\Rightarrow {\left(20+a^2\right)}^3=144\times 48\times a$
$\Rightarrow a=2$
$\Rightarrow$ Equation of plane is $2x+2y+4z=24$
Or $x+y+2z=12$
$\Rightarrow (3,0,4)$ Not lies on the Plane $x+y+2z=12$
So , the correct option is (B) : $(3,0,4)$

Answer 2

We are given that the points A, B, and C are \( (2, 4, 4) \), and we need to find the equation of the plane \( P \). 
Step 1: The equation of the plane can be written as: \[ \mathbf{r} = (2\hat{i} + 4\hat{j} + 4\hat{k}) \cdot \left[ (x - 2)\hat{i} + (y - 4)\hat{j} + (z - 4)\hat{k} \right] = 0. \] Step 2: Simplifying this expression, we get: \[ 2x + ay + 4z = 20 + a^2. \] Step 3: Substituting the coordinates of points A, B, and C: 
- For \( A = \left( \frac{20 + a^2}{2}, 0, 0 \right) \), 
- For \( B = \left( 0, \frac{20 + a^2}{a}, 0 \right) \), 
- For \( C = \left( 0, 0, \frac{20 + a^2}{4} \right) \). 
Step 4: We also know the volume of the tetrahedron is: \[ \text{Volume of tetrahedron} = \frac{1}{6} \left| \mathbf{a} \cdot \left( \mathbf{b} \times \mathbf{c} \right) \right| = 144. \] Step 5: From this, we find \( a = 2 \), and the equation of the plane becomes: \[ 2x + 2y + 4z = 24 \quad \Rightarrow \quad x + y + 2z = 12. \] Step 6: Now, to check if the point \( (3, 0, 4) \) lies on the plane: \[ x + y + 2z = 3 + 0 + 8 = 11 \quad (\text{not equal to } 12). \] Thus, \( (3, 0, 4) \) does not lie on the plane.