Three Dimensional Geometry Part 2

Given, point P(1,3,5) has the mirror image Q(α,β,γ) with respect to plane 4x−5y+2z=8 Then, M be the mid-point of line joining point P and Q. and M lies on given plane.
Coordinates of M(a,b,c) are as follows
a=‌

α+1

2

,b=‌

β+3

2

,c=‌

γ+5

2

∵ M lies on plane, then
‌4a−5b+2c=8
‌4(‌

α+1

2

)−5(‌

β+3

2

)+2(‌

γ+5

2

)=8. . . (i)
Also, PQ is perpendicular to plane
⇒‌‌‌

α−1

4

=‌

β−3

−5

=‌

γ−5

2

=λ‌ (say) ‌
⇒‌‌α=4λ+1,β=3−5λ,γ=2λ+5 . . . (ii)
Use Eq. (ii) in Eq. (i), we get
2(4λ+2)−5(‌

6−5λ

2

)+2λ+10=8
⇒‌‌8λ+4−15+‌

25λ

2

+2λ+10=8⇒λ=‌

2

5

∴‌‌α=4λ+1=4(‌

2

5

)+1=‌

13

5

‌‌β=3−5(‌

2

5

)=‌

5

5

=1
and ‌‌γ=5+2(‌

2

5

)=‌

29

5

∴5(α+β+γ)=5(‌

13

5

+1+‌

29

5

)=47