JEE MAIN - Mathematics (2022 - 28th July Morning Shift - No. 17)
Let $$\mathrm{P}(-2,-1,1)$$ and $$\mathrm{Q}\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$$ 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
450
Explanation
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d.r's of $$RS = < \alpha , - 1,\beta > $$
d.r's of $$PQ = < {{90} \over {17}},{{60} \over {17}},{{94} \over {17}} > \, = \, < 45,30,47 > $$
as PQ and RS are diagonals of rhombus
$$\alpha (45) + 30( - 1) + 47(\beta ) = 0$$
$$ \Rightarrow 45\alpha + 47\beta = 30$$
i.e., $$\alpha = {{30 - 47\beta } \over {45}}$$
for minimum integral value $$\alpha = - 15$$ and $$\beta = 15$$
$$ \Rightarrow {\alpha ^2} + {\beta ^2} = 450$$.
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