# Playing with co-ordinate geometry

Geometry Level 5

We have a plane in $$xyz$$-Cartesian Space whose equation is given by $$x+y+z=1$$.

I now take this plane as $${x}^{'}{y}^{'}$$-Cartesian Plane where $${x}^{'}$$ and $${y}^{'}$$-axes are given by $$x+y=1=z+1$$ and $$2x=2y=1-z$$ respectively.

The Direction cosines of $$\Large +{x}^{'}$$ and $$+{y}^{'}$$ axes are $$(\frac{-1}{\sqrt{2}},\frac{1}{\sqrt{2}},0)$$ and $$(\frac{-1}{\sqrt{6}},\frac{-1}{\sqrt{6}},\frac{2}{\sqrt{6}})$$ respectively.

A line is chosen in our given plane whose equation in $$xyz$$-Cartesian Space is given by :

$$3x-1=\frac{3y-1}{2}=\frac{3z-1}{-3}$$

If the equation of the above mentioned line in $${x}^{'}{y}^{'}$$-Cartesian Plane is given by :

$$a{x}^{'}+b{y}^{'}=1$$

Find $$\sqrt{2}a+\sqrt{6}b$$

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