If a point P( , , ) satisfying (gathered gathered)((|gathered 2 10 8 9 3 8 8 4 8 gathered)…

Question

If a point $P(\alpha ,\beta ,\gamma )$ satisfying

$(\begin{gathered} \alpha  \quad  \beta  \quad  \gamma \end{gathered})((|\begin{gathered} 2 \quad  10 \quad  8 \ 9 \quad  3 \quad  8 \ 8 \quad  4 \quad  8 \end{gathered}))|=(\begin{gathered} 0 \quad  0 \quad  0 \end{gathered})$

lies on the plane $2x+4y+3z=5$, then $6\alpha +9\beta +7\gamma $ is equal to :

Accepted Answer

Point $P(\alpha ,\beta ,\gamma )$ lies on the plane $2x+4y+3z=5$,

$	herefore $ $2\alpha +4\beta +3\gamma =5$ ........(1)

Given, $(\begin{gathered} \alpha  \quad  \beta  \quad  \gamma \end{gathered})((|\begin{gathered} 2 \quad  10 \quad  8 \ 9 \quad  3 \quad  8 \ 8 \quad  4 \quad  8 \end{gathered}))|=(\begin{gathered} 0 \quad  0 \quad  0 \end{gathered})$

$	herefore $ $2\alpha +9\beta +8\gamma =0$ .......(2)

and $10\alpha +3\beta +4\gamma =0$ ........(3)

and $8\alpha +8\beta +8\gamma =0$ ..........(4)

Subtract (4) from (2)

$-6\alpha +\beta =0$

$\beta =6\alpha $

From equation (4)

$8\alpha +48\alpha +8\gamma =0$

$\gamma =-7\alpha $

From equation (1)

$2\alpha +24\alpha -21\alpha =5$

$5\alpha =5$

$\alpha =1$

$\beta =+6,\, \gamma =-7$

$	herefore 6\alpha +9\beta +7\gamma $

$=6+54-49$

$=11$

Solutions · 1