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  4. The equation of the plane through the line of intersection of planes a x + b y + c z + d = 0, a' x + b' y + c' z + d'=0 and parallel to the line y =0, z =0 is

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Q. The equation of the plane through the line of intersection of planes $a x + b y + c z + d = 0, a' x + b' y + c' z + d'=0 $ and parallel to the line $y =0, z =0$ is

Three Dimensional Geometry

Solution:

Equation of a plane through the line of intersection of given plane is
$a x+b y+c z+d+\lambda\left(a' x+b' y+c' z+d'\right)=0$
$\Rightarrow \left(a+\lambda a'\right) x+\left(b+\lambda b'\right) y+\left(c+\lambda c'\right) z+\left(d+\lambda d'\right)=0$
It is parallel to $y =0, z =0$
i.e., $x$ -axis whose direction ratios are $1,0,0$
$\therefore 1\left(a+\lambda a'\right)+0\left(b+\lambda b'\right)+0\left(c+\lambda c'\right)=0$
$ \Rightarrow \lambda=-\frac{a}{a'}$
Hence, the required plane is
$y\left(a' b-a b'\right)+z\left(a' c-a c'\right)+\left(a' d-a d'\right)=0$