NDA Math Three Dimensional Geometry

Let a, b, c be the direction ratios of the line.
Then its equation is
‌‌

x−5

a

=‌‌

y+6

b

=‌‌

z−7

c

...(i)
Since (i) is parallel to the planes (x+y+z=1) and 2x−y−2z=3 then
a(1)+b(1)+c(1)=0 and a(z)+b(−1)+c(−2)=0
By cross multiplication
‌‌

a

−1

=‌‌

b

4

=‌‌

c

−3

=λ
⇒a=−λ,b=4λ,c=−3λ
⇒ Direction ratios of,the line are <−1,4,−3>=<1,−4,3>
Substituting a, b, c in (i), we get
‌‌

x−5

−1

=‌‌

y+6

4

=‌‌

z−7

−3

...(ii)