Explanation
Any plane through \left( 1,0,0 \right) is
A\left( x-1 \right) +B\left( y-0 \right) +C\left( z-0 \right) =0 ...(1)
It contains \left( 0,1,0 \right) if -A+B=0 ....(2)
Also, (1) makes an angle of \displaystyle \dfrac { \pi }{ 4 } with the plane x+y=3
Therefore, \displaystyle \cos { \dfrac { \pi }{ 4 } } =\dfrac { \left| A+B \right| }{ \sqrt { { A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 } } \sqrt { { 1 }^{ 2 }+{ 1 }^{ 2 } } }
\Rightarrow { \left( A+B \right) }^{ 2 }={ A }^{ 2 }+{ B }^{ 2 }+{ C }^{ 2 }\Rightarrow 2AB={ C }^{ 2 } ....(3)
From (2) and (3), { C }^{ 2 }=2{ A }^{ 2 }\Rightarrow C=\pm \sqrt { 2 } A
Hence, A:B:C::A:A:\pm \sqrt { 2 } A
\therefore direction ratios are 1:1:\pm \sqrt { 2 }
Eliminating l from equations n=l+m, \; m=2l+3n, we get
\dfrac{m}{n}= \dfrac{5}{3}...(1)
Similarly, eliminating m from the two equations we get
\dfrac{n}{l}=\dfrac{3}{-2}...(2)
From (1) and (2), m:n:l = 5:3:-2
Let m=5t,\; n=3t,\; l=-2t
Now, use l^2+m^2+n^2=1
\Rightarrow (25+9+4)t^2=1
\Rightarrow t^2= \dfrac{1}{38}
\Rightarrow t= \pm \dfrac{1}{\sqrt{38}}
Positive value of t will give direction cosines of one line and negative value will give direction cosines of another line.
Hence, they will be parallel to each other with angle between them 180^{o}
Please disable the adBlock and continue. Thank you.