Mathematics - Quadratic Equation Question with Solution | TestHub

We have,

$x+y+z=5$

$x^2+y^2+z^2=9$

$\Rightarrow x^2+y^2+(5-x-y{)}^2=9$

$\Rightarrow y^2+x^2-5x-5y+xy+8=0$

$\Rightarrow y^2+y(x-5)+\left(x^2-5x+8\right)=0$

$\Rightarrow$ Since $y\in R$

$(x-5{)}^2-4\left(x^2-5x+8\right)\ge 0$

$\Rightarrow x^2-10x+25-4x^2+20x-32\ge 0$

$\Rightarrow 3x^2-10x+7\le 0$

$\Rightarrow 3x^2-7x-3x+7\le 0$

$\Rightarrow (3x-7)(x-1)\le 0$

$\Rightarrow x\in \left[1,\frac{7}{3}\right]$
Length of interval $=\left(\frac{7}{3}-1\right)=\frac{4}{3}$