Question

If the given planes $ax+by+cz+d=0$ and $ax+by+cz+d=0$ be mutually perpendicular, then

• A. $\frac{a}{a}=\frac{b}{b}=\frac{c}{c}$
• B. $\frac{a}{a}+\frac{b}{b}+\frac{c}{c}=0$
• C. $aa+bb+cc+dd=0$
• D. $aa+bb+cc=0$

Answer 1

Answer: (D)

$aa+bb+cc=0$

$ax+by+cz+d=0$ and $a^{1}x+b^{1}y+c^{1}z+d^1=0$ are mutually perpendicular then their respective norm also will be perpendicular too.

Hence by the perpendicular condition,

${aa}^1+{bb}^1+{cc}^1=0$

Where, $⟨a,b,c⟩$ are the direction ratio of the normal to the plane $ax+by+cz+d=0$ and $⟨a^1,b^1,c^1⟩$ are the direction ratio of the normal to the plane $a^{1}x+b^{1}y+c^{1}z+d^1=0$

Correct option will be $\left(D\right)$

(But there's a formating error. In the question both plane equation are same, which contradicts the fact that the planes are perpendicular).