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 lines $y=0=z$ is

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

Answer 1

Answer: (D)

None of these

Equation of planes passes through the intersection of the planes $ax+by+cz+d=0$ and $a^{\prime }x+b^{\prime }y+c^{\prime }z+d=0$ is
$(a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime })+\lambda (ax+by+cz+d)=0$...(1)
which is parallel to $y=0=z$ means parallel to $x$-axis.
$\therefore (a^{\prime }+a\lambda )1+0(b^{\prime }+b\lambda )+0(c^{\prime }+c\lambda )=0$
$\therefore a\lambda =a^{\prime }$
$\therefore \lambda =-\frac{a^{\prime }}{a}$
Putting $\lambda =\frac{a^{\prime }}{a}$ in (1), we have
$\therefore a(a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime })-a^{\prime }(ax+by+(z+d\left)\right)=0$
$\Rightarrow y(a^{\prime }b-a{b}^{\prime })+z(a^{\prime }c-a{c}^{\prime })+a^{\prime }d-a{d}^{\prime }=0$