Q.The plane ____ passes through the line of intersection of the planes $x+y=0$ and $x-y=0$. Choices given: $[x=0,\ x+2y=0,\ x-2y=0,\ y+2x=0]$

Any plane through the line of intersection of $P_1=0$ and $P_2=0$ has the form $P_1+\lambda P_2=0$; here the two given planes both pass through the origin along the $Z$-axis, so technically every plane $px+qy=0$ contains that line — the natural, most direct member (taking $\lambda=1$, i.e. adding the two equations) is singled out below.

Family of planes through the intersection of $x+y=0$ and $x-y=0$: $(x+y)+\lambda(x-y)=0$, i.e. $(1+\lambda)x+(1-\lambda)y=0$.

Since both given planes pass through the origin and their line of intersection is the $Z$-axis ($x=0,y=0$), every plane of the form $px+qy=0$ contains this line — so strictly speaking more than one of the listed options ($x=0$, $x+2y=0$, $x-2y=0$, $y+2x=0$) technically belongs to the family for some value of $\lambda$.

…