Question

The reflection of the plane $a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime }=0$ in the plane $ax+by+cz+d=0$, is given by the relation

• A. $\frac{a^2+b^2+c^2}{{aa}^{\prime }+{bb}^{\prime }+{cc}^{\prime }}=\frac{2({ax}^{\prime }+{by}^{\prime }+{cz}^{\prime }+d^{\prime })}{ax+by+cz+d}$
• B. $\frac{a^2+b^2+c^2}{{aa}^{\prime }+{bb}^{\prime }+{cc}^{\prime }}=\frac{2(ax+by+cz+d)}{{ax}^{\prime }+{by}^{\prime }+{cz}^{\prime }+d^{\prime }}$
• C. $\frac{2({aa}^{\prime }+{bb}^{\prime }+{cc}^{\prime })}{a^2+b^2+c^2}=\frac{{ax}^{\prime }+{by}^{\prime }+{cz}^{\prime }+d^{\prime }}{ax+by+cz+d}$
• D. None of these

Answer 1

Answer: (B), (C)

$\frac{a^2+b^2+c^2}{{aa}^{\prime }+{bb}^{\prime }+{cc}^{\prime }}=\frac{2(ax+by+cz+d)}{{ax}^{\prime }+{by}^{\prime }+{cz}^{\prime }+d^{\prime }}$
$\frac{2({aa}^{\prime }+{bb}^{\prime }+{cc}^{\prime })}{a^2+b^2+c^2}=\frac{{ax}^{\prime }+{by}^{\prime }+{cz}^{\prime }+d^{\prime }}{ax+by+cz+d}$

We know that he reflection (image) of the plane
$p_1:a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime }=0$ in the plane
$p_2:ax+by+cz+d=0$ is given by

$2(a{a}^{\prime }+b{b}^{\prime }+c{c}^{\prime })(ax+by+cz+d)=(a^2+b^2+c^2)(a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime })$

$\Rightarrow \frac{a^2+b^2+c^2}{{aa}^{\prime }+{bb}^{\prime }+{cc}^{\prime }}=\frac{2(ax+by+cz+d)}{{ax}^{\prime }+{by}^{\prime }+{cz}^{\prime }+d^{\prime }}$
$\Rightarrow \frac{2({aa}^{\prime }+{bb}^{\prime }+{cc}^{\prime })}{a^2+b^2+c^2}=\frac{{ax}^{\prime }+{by}^{\prime }+{cz}^{\prime }+d^{\prime }}{ax+by+cz+d}$
Hence option B,C are correct.