1. Key Concepts and Formulas

• General Equation of a Plane: The equation of any plane in 3D space can be represented as $Ax + By + Cz + D = 0$, where $(A, B, C)$ is the normal vector to the plane and $D$ is a constant.
• Plane Containing an Axis: If a plane contains one of the coordinate axes (e.g., the y-axis), it implies two crucial conditions:
  1. The plane must pass through the origin $(0,0,0)$. This means that when $(0,0,0)$ is substituted into the plane equation, the equation must hold, which simplifies to $D=0$.
  2. The normal vector to the plane, $\vec{n} = (A,B,C)$, must be perpendicular to the direction vector of the axis it contains. For the y-axis, its direction vector is $\vec{j} = (0,1,0)$. The dot product of perpendicular vectors is zero, so $\vec{n} \cdot \vec{j} = 0$.
  • Combining these for a plane containing the y-axis, the equation simplifies to $Ax + Cz = 0$.
• Plane Passing Through a Point: If a plane passes through a specific point $(x_0, y_0, z_0)$, then substituting these coordinates into the plane's equation must satisfy the equation.

2. Step-by-Step Solution

Let the general equation of the plane be $Ax + By + Cz + D = 0$.

• Step 1: Apply the condition that the plane contains the y-axis.

  • Why this step? This condition simplifies the general equation of the plane by determining some of its coefficients, making it easier to solve for the remaining ones.
  • Since the plane contains the y-axis, it must pass through the origin $(0,0,0)$. Substituting $(0,0,0)$ into the general plane equation: $A(0) + B(0) + C(0) + D = 0 \implies D = 0$ So, the equation simplifies to $Ax + By + Cz = 0$.
  • Additionally, if the plane contains the y-axis, its normal vector $\vec{n} = (A,B,C)$ must be perpendicular to the direction vector of the y-axis, $\vec{j} = (0,1,0)$. The dot product of perpendicular vectors is zero: $\vec{n} \cdot \vec{j} = (A,B,C) \cdot (0,1,0) = 0$ $A(0) + B(1) + C(0) = 0 \implies B = 0$
  • Result: Combining $D=0$ and $B=0$, any plane containing the y-axis must have an equation of the form $Ax + Cz = 0$.

• Step 2: Apply the condition that the plane passes through the point (1, 2, 3).

  • Why this step? This condition allows us to establish a relationship between the remaining unknown coefficients, $A$ and $C$, by utilizing the given point.
  • Since the plane passes through the point $(1, 2, 3)$, these coordinates must satisfy the simplified plane equation $Ax + Cz = 0$. Substitute $x=1$ and $z=3$ into the equation (note that the y-coordinate $y=2$ is not used here because $B=0$): $A(1) + C(3) = 0$ $A + 3C = 0$

• Step 3: Determine the final equation of the plane.

  • Why this step? We now have a relationship between $A$ and $C$, which allows us to write the final equation of the plane in its simplest form.
  • From the relation $A + 3C = 0$, we can express $A$ in terms of $C$: $A = -3C$
  • Substitute this expression for $A$ back into the simplified plane equation $Ax + Cz = 0$: $(-3C)x + Cz = 0$
  • Factor out $C$ from the equation: $C(-3x + z) = 0$
  • Since $C$ cannot be zero (if $C=0$, then from $A=-3C$, $A$ would also be $0$, leading to a normal vector $(0,0,0)$ which does not define a plane), we can divide the entire equation by $C$: $-3x + z = 0$
  • Rearranging the terms, we get the equation of the plane: $3x - z = 0$

3. Common Mistakes & Tips

• Missing the $D=0$ condition: A frequent error is to forget that a plane containing any axis must pass through the origin, thus eliminating the constant term $D$.
• Incorrectly handling the normal vector: Remember that the normal vector to the plane must be perpendicular to the direction vector of the axis it contains. This directly sets the corresponding coefficient to zero (e.g., $B=0$ for the y-axis).
• Algebraic errors: Be meticulous with substitutions and algebraic manipulations to avoid simple calculation mistakes that can lead to an incorrect final equation.

4. Summary

To find the equation of the plane, we first used the condition that it contains the y-axis to simplify its general form to $Ax + Cz = 0$. Next, we utilized the given point $(1, 2, 3)$ to establish a relationship between the coefficients $A$ and $C$ as $A + 3C = 0$. Finally, by substituting $A = -3C$ back into the simplified plane equation and dividing by $C$, we obtained the equation of the plane as $3x - z = 0$.

5. Final Answer The final answer is $\boxed{x + 3z = 0}$. This corresponds to option (A).