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The acute angle between the planes \mathrm{P_{1}} and \mathrm{P_{2},} when \mathrm{P_{1}} and \mathrm{P_{2}} are  the planes passing through the intersection of the planes \mathrm{5x+8y+13z-29= 0} and \mathrm{8x-7y+z-20= 0} and the points \mathrm{\left ( 2,1,3 \right )} and \mathrm{\left ( 0,1,2 \right )}, respectively, is

Option: 1

\frac{\pi}{3}


Option: 2

\frac{\pi}{4}


Option: 3

\frac{\pi}{6}


Option: 4

\frac{\pi}{12}


Answers (1)

best_answer

\mathrm{Let \: P_{1}:(5 x+2 y+13 z-29)+\lambda(8 x-7 y+z-20)=0}
Passes through \mathrm{(2,1,3)}
\mathrm{\Rightarrow(10+8+39-29)+\lambda(16-7+3-20)=0 \Rightarrow \lambda=\frac{7}{2}}
\mathrm{\Rightarrow P_{1} :(10 x+16 y+26 z-58)+(56 x-49 y+73-140)=0}
\mathrm{\Rightarrow 66 x-33 y+33 z-198=0 \Rightarrow 2 x-y+z-6=0\: ---(1)}


\mathrm{Let\: P_{2}:(5 x+8 y+13 z-29)+\mu(8 x-7 y+z-20)=0}
Passes through \mathrm{(0,1,2)}
\mathrm{\Rightarrow(0+8+26-29)+\mu(0-7+2-20)=0 \Rightarrow \mu=1 / 5}
\mathrm{\Rightarrow P_{2}:(25 x+40 y+65 z-145)+8 x-7 y+z-20=0 \Rightarrow 33 x+33 y+663-165=0}
\mathrm{\Rightarrow x+y+2z-5= 0\, ---(2)}
\mathrm{\cos \theta=\left|\frac{(2,-1,1) \cdot(1,1,2)}{\sqrt{2^{2}+1^{2}+1^{2}} \sqrt{1^{2}+1^{2}+2^{2}}}\right|=\frac{3}{6}=\frac{1}{2} \Rightarrow \theta=\pi / 3}

Option (A)

Posted by

seema garhwal

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