Question

The equation of the plane which contains the $y$-axis and passes through the point $\left(1,2,3\right)$ is:

• A. $3x+z=6$
• B. $3x-z=0$
• C. $x+3z=10$
• D. $x+3z=0$

Answer 1

The correct option is B

$3x-z=0$

Explanation for the correct option:

Step 1: Write the equation of a plane carrying a point

Equation of a plane is given as

$\mathbf{a}\left(x-x_1\right)\mathbf{+}\mathbf{b}\left(y-y_1\right)\mathbf{+}\mathbf{c}\left(z-z_1\right)\mathbf{=}\mathbf{0}$

We have $x_1=1,y_1=2,z_1=3$

Step 2: Use the direction ratios of $y-$axis to find $b$

The direction ratios of the $y-$axis are $0,1,0$

$\Rightarrow$ $0.a+1.b+0.c=0$

$\Rightarrow$ $b=0$

Step 3: Find the relation between $a$ and $c$

So, the equation of plane is

$a\left(x-1\right)+c(z-3)=0$

$x=0,z=0$ also satisfy this equation

$\Rightarrow$ $a+3c=0$

$\Rightarrow$ $a=-3c$

Step 4: Write the equation of the plane

So, the equation of the plane is given as

$-3c(x-1)+c(z-3)=0$

$\Rightarrow$ $-3x+3+z-3=0$

$\Rightarrow$ $3x-z=0$ is the equation of the plane.

Hence, option (B) is the correct answer.