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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)

best_answer

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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