If the mirror image of the point (1,3,5) with respect to the plane $4x -5y +2z=8$ is $(\alpha ,\beta ,\gamma )$ then $5(\alpha ,\beta ,\gamma )$ equals: Option: 1 47 Option: 2 39 Option: 3 41 Option: 4 43

Point Q is image of point P with respect  to the plane, M is the midpoint of P and Q, lies in plane

$\mathrm{M}\left(\frac{1+\alpha}{2}, \frac{3+\beta}{2}, \frac{5+\gamma}{2}\right)$

this point lies in the plane 4x – 5y + 2z = 8

So,

$\\4\left(\frac{1+\alpha}{2}\right)-5\left(\frac{3+\beta}{2}\right)+2\left(\frac{5+\gamma}{2}\right)=8\\$

Also PQ perpendicualr to the plane

$\\\Rightarrow \overrightarrow{\mathrm{PQ}} \| \overrightarrow{\mathrm{n}} \\ \frac{\alpha-1}{4}=\frac{\beta-3}{-5}=\frac{\gamma-5}{2}=\mathrm{k} \text { (let) } \\\alpha=4k+1,\;\beta=-5k+3,\;\gamma=2k+5$

$\\2(1+4 k)-5\left(\frac{6-5 k}{2}\right)+(10+2 k)=8 \\ k=\frac{2}{5}$

$\\ \alpha=\frac{13}{5}, \beta=1, \gamma=\frac{29}{5}\\ 5(\alpha+\beta+\gamma)=13+5+29=47$

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