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 α2+β2 is equal to

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 α2+β2 is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

RS ≡(α,-1, β) DR of PQ ≡((56/17)+2, (43/17)+1, (111/17)-1) ≡((90/17), (60/17), (94/17)) (90/17) α+(60/17)(-1)+(94/17) β=0 90 α+94 β=60 β=(60-90 α/94) β=(30(2-3 α)/94) β=-30 ((3 α-2)/94) β=(-15/47)(3 α-2) ⇒ (β/-15)=(3 α-2/47) ⇒ β=-15, α=-15 α2+β2=225+225 =450

Q. Let P(−2,−1,1) and Q(1756​,1743​,17111​) 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 α2+β2 is equal to ___

RS≡(α,−1,β) DR of PQ≡(1756​+2,1743​+1,17111​−1) ≡(1790​,1760​,1794​) 1790​α+1760​(−1)+1794​β=0 90α+94β=60 β=9460−90α​ β=9430(2−3α)​ β=−3094(3α−2)​ β=47−15​(3α−2) ⇒−15β​=473α−2​ ⇒β=−15,α=−15 α2+β2=225+225 =450

Q. Let $P (- 2, - 1, 1)$ and $Q (\frac{56}{17}, \frac{43}{17}, \frac{111}{17})$ 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 ___