1. Family of planes through the line of intersection

If a plane passes through the line of intersection of $P_1: x+2y+3z=2$ and $P_2: x-y+z=3,$ then its equation is of the form $P_1+\lambda P_2=0,$ i.e. $(x+2y+3z-2)+\lambda(x-y+z-3)=0.$

So the required plane is $(1+\lambda)x+(2-\lambda)y+(3+\lambda)z-(2+3\lambda)=0.$

2. Use the distance condition

The distance of point $(3,1,-1)$ from the plane $ax+by+cz+d=0$ is $\frac{|a x_1+b y_1+c z_1+d|}{\sqrt{a^2+b^2+c^2}}.$

Here, $a=1+\lambda,\quad b=2-\lambda,\quad c=3+\lambda,\quad d=-(2+3\lambda).$

Substitute $(3,1,-1)$: [ (1+\lambda)\cdot 3+(2-\lambda)\cdot 1+(3+\lambda)(-1)-(2+3\lambda). ] Simplify: [ 3+3\lambda+2-\lambda-3-\lambda-2-3\lambda=-2\lambda. ] Thus numerator is $|-2\lambda|=2|\lambda|.$

Now denominator: [ \sqrt{(1+\lambda)^2+(2-\lambda)^2+(3+\lambda)^2}. ] Expand: [ (1+\lambda)^2=1+2\lambda+\lambda^2, ] [ (2-\lambda)^2=4-4\lambda+\lambda^2, ] [ (3+\lambda)^2=9+6\lambda+\lambda^2. ] Adding, [ 1+4+9+(2\lambda-4\lambda+6\lambda)+3\lambda^2 =14+4\lambda+3\lambda^2. ] So distance is $\frac{2|\lambda|}{\sqrt{3\lambda^2+4\lambda+14}}.$

Given distance is $\frac{2}{\sqrt{3}}.$ Hence, $\frac{2|\lambda|}{\sqrt{3\lambda^2+4\lambda+14}}=\frac{2}{\sqrt{3}}.$

Cancel 2: $\frac{|\lambda|}{\sqrt{3\lambda^2+4\lambda+14}}=\frac{1}{\sqrt{3}}.$

Square both sides: $\frac{\lambda^2}{3\lambda^2+4\lambda+14}=\frac13.$

Therefore, $3\lambda^2=3\lambda^2+4\lambda+14.$ So, $4\lambda+14=0\quad\Rightarrow\quad \lambda=-\frac72.$

3. Substitute back into plane equation

Using $(1+\lambda)x+(2-\lambda)y+(3+\lambda)z-(2+3\lambda)=0,$ put $\lambda=-\dfrac72$:

• $1+\lambda=1-\dfrac72=-\dfrac52$
• $2-\lambda=2+\dfrac72=\dfrac{11}{2}$
• $3+\lambda=3-\dfrac72=-\dfrac12$
• $2+3\lambda=2-\dfrac{21}{2}=-\dfrac{17}{2}$

So the plane is $-\frac52 x+\frac{11}{2}y-\frac12 z-\left(-\frac{17}{2}\right)=0.$ That is, $-\frac52 x+\frac{11}{2}y-\frac12 z+\frac{17}{2}=0.$ Multiply by $2$: $-5x+11y-z+17=0.$ Or, $5x-11y+z=17.$

4. Match with options

This is exactly Option A.

5. Comparison with stored answer

Stored correct answer: A

Derived answer: A

They match.