Question

Let P(–2, –1, 1) and Q(56/17, 43/17, 111/17) be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are $α, –1, β,$ where both $α$ and $β$ are integers of minimum absolute values, then $α^2 + β^2$ is equal to ___________.

Answer 1

$RS≡(α,−1,β)$

$DR \space of\space PQ$≡$(\frac{56}{17}+2,\frac{43}{17}+1,\frac{111}{17}−1)$
≡$(\frac{90}{17},\frac{60}{17},\frac{94}{17})$

$\frac{90}{17}α+\frac{60}{17}(−1)+\frac{94}{17}β=0$

$90α+94β=60$
$β=\frac{60−90α}{94}$

$β=\frac{30(2−3α)}{94}$

$β=−30\frac{(3α−2)}{94}$

$β=−\frac{15}{47}(3α−2)$

$⇒\frac{β}{-15}=\frac{3α−2}{47}$

$⇒β=−15,α=−15$

$α^2+β^2=225+225$

$=450$