જો સમીકરણ સંહિત 2 x+3 y-z=5 | vedclass

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Question

Using family of planes

$2 \mathrm{x}+3 \mathrm{y}-\mathrm{z}-5=\mathrm{k}_1(\mathrm{x}+\alpha \mathrm{y}+3 \mathrm{z}+4)+\mathrm{k}_2(3 \mathrm{x}-

Vedclass · Accepted Answer

$1120$

Vedclass · Answer

Using family of planes

$2 \mathrm{x}+3 \mathrm{y}-\mathrm{z}-5=\mathrm{k}_1(\mathrm{x}+\alpha \mathrm{y}+3 \mathrm{z}+4)+\mathrm{k}_2(3 \mathrm{x}-\mathrm{y}+\beta \mathrm{z}-7)$

$2=\mathrm{k}_1+3 \mathrm{k}_2, 3=\mathrm{k}_1 \alpha-\mathrm{k}_2,-1=3 \mathrm{k}_1+\beta \mathrm{k}_2,-5=4{k}_1-7 \mathrm{k}_2$

On solving we get

$\begin{gathered}k_2=\frac{13}{19}, k_1=\frac{-1}{19}, \alpha=-70, \beta=\frac{-16}{13} \ 13 \alpha \beta=13(-70)\left(\frac{-16}{13}ight) \ =1120\end{gathered}$

જો સમીકરણ સંહિત $2 x+3 y-z=5$ ; $x+\alpha y+3 z=-4$ ; $3 x-y+\beta z=7$ ને અસંખ્ય ઉકેલો હોય, તો $13 \alpha \beta$=____________.

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