A plane passing through the points \left ( 0,-1,0 \right ) and \left ( 0,0,1 \right ) and making an angle \frac{\pi }{4} with the plane y-z+5=0, also passes through the point :

  • Option 1)

       \left ( -\sqrt{2},1,-4 \right )          

  • Option 2)

     \left ( \sqrt{2},-1,4 \right )

  • Option 3)

     \left (- \sqrt{2},-1,-4 \right )   

  • Option 4)

     \left ( \sqrt{2},1,4 \right )

 

Answers (1)

Let eq. of plane is   a\left ( x-0 \right )+b\left ( y+1 \right )+c\left ( z-0 \right )=0

  \Rightarrow ax +by+cz+b=0

as given it passes through \left ( 0,0,1 \right )

            c=-b

\therefore equation of plane is

ax+by-bz+bc=0

Also given that this plane make \frac{\pi }{4} with the plane y-z+5=0

cos\frac{\pi }{4}=\frac{o+b+b}{\sqrt{a^{2}+b^{2}+b^{2}}\sqrt{2}}=\frac{1}{\sqrt{2}}

     a=\pm \sqrt{2}b

Now eq.of plane become 

\pm \sqrt{2}bx+by-bz+b=0

\Rightarrow \pm \sqrt{2}x+y-z+1=0

  \left ( \sqrt{2},1,4 \right ) Satisfies

 


Option 1)

   \left ( -\sqrt{2},1,-4 \right )          

Option 2)

 \left ( \sqrt{2},-1,4 \right )

Option 3)

 \left (- \sqrt{2},-1,-4 \right )   

Option 4)

 \left ( \sqrt{2},1,4 \right )

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