Question

Let $P (-2,-1,1)$ and $Q \left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$ be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $\alpha,-1, \beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then $\alpha^2+\beta^2$ is equal to ___

Answer 1

$RS \equiv(\alpha,-1, \beta)$
$DR$ of $PQ \equiv\left(\frac{56}{17}+2, \frac{43}{17}+1, \frac{111}{17}-1\right)$
$ \equiv\left(\frac{90}{17}, \frac{60}{17}, \frac{94}{17}\right) $
$\frac{90}{17} \alpha+\frac{60}{17}(-1)+\frac{94}{17} \beta=0 $
$ 90 \alpha+94 \beta=60 $
$ \beta=\frac{60-90 \alpha}{94}$
$ \beta=\frac{30(2-3 \alpha)}{94} $
$ \beta=-30 \frac{(3 \alpha-2)}{94} $
$ \beta=\frac{-15}{47}(3 \alpha-2) $
$\Rightarrow \frac{\beta}{-15}=\frac{3 \alpha-2}{47} $
$ \Rightarrow \beta=-15, \alpha=-15 $
$ \alpha^2+\beta^2=225+225 $
$ =450$