Question

Solve the following equations :
$x(x+y+z)=a^2,y(x+y+z)=b^2,z(x+y+z)=c^2.$

• A. $x=\pm \frac{a}{\surd [a^2+b^2+c^2]}$
  $y=\pm \frac{b}{\surd [a^2+b^2+c^2]}$
  $z=\pm \frac{c}{\surd [a^2+b^2+c^2]}$.
• B. $x=\pm \frac{a^2}{\surd [a^2+b^2+c^2]}$
  $y=\pm \frac{b^2}{\surd [a^2+b^2+c^2]}$
  $z=\pm \frac{c^2}{\surd [a^2+b^2+c^2]}$.
• C. $x=\pm \frac{a^3}{\surd [a^2+b^2+c^2]}$
  $y=\pm \frac{b^3}{\surd [a^2+b^2+c^2]}$
  $z=\pm \frac{c^3}{\surd [a^2+b^2+c^2]}$.
• D. None

Answer 1

Answer: (B)

$x=\pm \frac{a^2}{\surd [a^2+b^2+c^2]}$

$y=\pm \frac{b^2}{\surd [a^2+b^2+c^2]}$
$z=\pm \frac{c^2}{\surd [a^2+b^2+c^2]}$.

$x(x+y+z)=a^2........\left(1\right)y(x+y+z)=b^2........\left(2\right)z(x+y+z)=c^2........\left(3\right)$

Adding these 3 equations

$(x+y+z)(x+y+z)=a^2+b^2+c^2\Rightarrow (x+y+z{)}^2=a^2+b^2+c^{2}x+y+z=\pm \sqrt{a^2+b^2+c^2}$

From (1) $x=\frac{a^2}{\pm \sqrt{a^2+b^2+c^2}}=\pm \frac{a^2}{\sqrt{a^2+b^2+c^2}}$

Similarly

$y=\pm \frac{b^2}{\surd [a^2+b^2+c^2]}$

$z=\pm \frac{c^2}{\surd [a^2+b^2+c^2]}$.