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  • If the mirror image of the point (1,3,5) with respect to the plane  is <img alt="(\alpha ,\beta ,\gamma )" src="https://entrancecor

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

Answers (1)

best_answer

 

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

Posted by

himanshu.meshram

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