Question

Let ${\sigma }_1,{\sigma }_2,{\sigma }_3$ be planes passing through the origin. Assume that ${\sigma }_1$ is perpendicular to the vector $(1,1,1)$, ${\sigma }_2$ is perpendicular to a vector $(a,b,c)$, and ${\sigma }_3$ is perpendicular to the vector $(a^2,b^2,c^2)$. What are all the positive values of $a,b$, and $c$ so that ${\sigma }_1\cap {\sigma }_2\cap {\sigma }_3$ is a single point?

• A. Any positive value of $a,b,$ and $c$ other than $1$
• B. Any positive values of $a,b,$ and $c$ where either $a\ne b,b\ne c$ or $a\ne c$
• C. Any three distinct positive values of $a,b,$ and $c$
• D. There exist no such positive real numbers $a,b,$ and $c$

Answer 1

The correct option is C Any three distinct positive values of $a,b,$ and $c$

${\sigma }_1$ can be written as,
$x+y+z=0$
Similarly other two planes can be written as,
${\sigma }_2:ax+by+cz=0{\sigma }_3:a^{2}x+b^{2}y+c^{2}z=0$
For unique solution of the three equations,
$\left|\begin{array}{ccc}1& 1& 1\\ a& b& c\\ a^2& b^2& c^2\end{array}\right|\ne 0$
Using column operations as follows
$C_2\to C_2-C_1$ and $C_2\to C_3-C_1$
$\left|\begin{array}{ccc}1& 0& 0\\ a& b-a& c-a\\ a^2& b^2-a^2& c^2-a^2\end{array}\right|\ne 0\Rightarrow (c^2-a^2)(b-a)-(b^2-a^2)(c-a)\ne 0\Rightarrow (c-a)(b-a)(c+a-(b+a\left)\right)\ne 0\Rightarrow (c-a)(b-a)(c-b)\ne 0\Rightarrow (a-b)(b-c)(c-a)\ne 0$
All the three numbers should be distinct.