Question

The equation of the plane through the line of intersection of planes $ax+by+cz+d=0,a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime }=0$ and parallel to the line $y=0,z=0$ is

• A. $\left(a{b}^{\prime }-a^{\prime }b\right)x+\left(b{c}^{\prime }-b^{\prime }c\right)y+\left(a{d}^{\prime }-a^{\prime }d\right)=0$
• B. $\left(a{b}^{\prime }-a^{\prime }b\right)x+\left(b{c}^{\prime }-b^{\prime }c\right)y+\left(a{d}^{\prime }-a^{\prime }d\right)z=0$
• C. $\left(a{b}^{\prime }-a^{\prime }b\right)y+\left(a{c}^{\prime }-a^{\prime }c\right)z+\left(a{d}^{\prime }-a^{\prime }d\right)=0$
• D. None of these

Answer 1

Answer: (C)

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