Find the equation of the plane passing through the intersection of the planes $2x + 3y - z + 1 = 0$ and $x + y - 2z + 3 = 0$ and perpendicular to the plane $3x - y - 2z - 4 = 0$.

**OR**

Find the image of the point $(1, 2, 3)$ in the plane $x + 2y + 4z = 38$.

Question

Find the equation of the plane passing through the intersection of the planes $2x + 3y - z + 1 = 0$ and $x + y - 2z + 3 = 0$ and perpendicular to the plane $3x - y - 2z - 4 = 0$.

**OR**

Find the image of the point $(1, 2, 3)$ in the plane $x + 2y + 4z = 38$.

Accepted Answer

>

> [!TLDR]
> Write the family of planes through the given line of intersection with a parameter $k$, then fix $k$ by requiring the normal to be perpendicular to the third plane's normal.
>
> The plane through the intersection of $2x+3y-z+1=0$ and $x+y-2z+3=0$ can be written as
> $$(2x+3y-z+1)+k(x+y-2z+3)=0$$
> $$(2+k)x+(3+k)y+(-1-2k)z+(1+3k)=0 \qquad (\ast)$$
>
> Its normal is $\vec n_1=(2+k,\ 3+k,\ -1-2k)$. The given plane $3x-y-2z-4=0$ has normal $\vec n_2=(3,-1,-2)$.
>
> For $(\ast)$ to be perpendicular to this plane, $\vec n_1\cdot\vec n_2=0$:
> $$3(2+k)+(-1)(3+k)+(-2)(-1-2k)=0$$
> $$(6+3k)+(-3-k)+(2+4k)=0$$
> $$5+6k=0\ \Rightarrow\ k=-\dfrac56$$
>
> Substitute back into $(\ast)$:
> $$2+k=2-\dfrac56=\dfrac76,\quad 3+k=3-\dfrac56=\dfrac{13}6,\quad -1-2k=-1+\dfrac53=\dfrac23,\quad 1+3k=1-\dfrac52=-\dfrac32$$
>
> $$\dfrac76 x+\dfrac{13}6 y+\dfrac23 z-\dfrac32=0$$
>
> Multiply through by $6$:
> $$7x+13y+4z-9=0$$
>
> **Check:** normal $(7,13,4)\cdot(3,-1,-2)=21-13-8=0$ ✓ — perpendicular to $3x-y-2z-4=0$, and it belongs to the family so it passes through the line of intersection of the first two planes.
> …

Q.Find the equation of the plane passing through the intersection of the planes $2x + 3y - z + 1 = 0$ and $x + y - 2z + 3 = 0$ and perpendicular to the plane $3x - y - 2z - 4 = 0$ . OR Find the image of the point $(1, 2, 3)$ in the plane $x + 2y + 4z = 38$ .

Concept understanding — Equation of a Plane

Equation of a Plane

Point + normal form

Normal (perpendicular) form

Intercept form …

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