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(a) Let the direction atios of required line he fegportional to 9,6, c as it lies on 60^{\text {th }} Planes, so their normals are \perp to it.Given,\begin{array}{l}P_{1}=x+2 y-z=2 \\P_{2}=2 x-y+4 z=5\end{array}using cross muctiply,\frac{a}{7}=\frac{b}{-2-4}=\frac{c}{-1-4}=\frac{a}{7}=\frac{b}{-6}=\frac{c}{-5}lini frassing through origin sits clifection ratics are proportionalto7,-6,-5so, eq'n of lim:-=\frac{x}{7}=\frac{4}{-6}=\frac{2}{-5}(6) Equation of Llani:-\begin{array}{l}m_{1}=x+2 y-z=2 \\m_{2}=2 x-y+4 z=5 \\\Rightarrow m_{1}+\lambda m_{2} \\\Rightarrow x+2 y-z-2+\lambda(2 x-y+4 z-5)=0 \\\Rightarrow \quad x+2 y-z-2+2 \lambda x-y \lambda+4 z \lambda-5 \lambda=0 \\\Rightarrow(1+2 \lambda) x+(2-\lambda) y+(-1+4 \lambda) z=0 \\\Rightarrow \text { eq'h Passing to foint }=(1,1,1) \\\Rightarrow \delta_{1}=-4+\lambda(0)=0 \\\lambda=0 \\\text { So, } g^{\prime}-i 4=x+2 y-z-2=0 \text { As } \\\end{array} ...