Q100 / 160 · TS EAMCET 2019 (03 May Shift 1) · Mathematics

If a plane $P$ passes through the points $(1,0,0),(0,1,0)$ and makes an angle $\frac{\pi}{4}$ with the plane $x+y=3$, then the direction ratios of a normal to that plane $P$ is

  • A
    $1, \sqrt{2}, 1$
  • Correct answer
    B
    $1,1, \sqrt{2}$
  • C
    $1,1,2$
  • D
    $\sqrt{2}, 1,1$

Solution

Let the direction ratio of normal to the plane $P$ is $(a, b, c)$. $\because$ equation of another given plane $x+y=3$ having direction ratio of normal $=(1,1,0)$ $$ \begin{aligned} & \therefore \cos \frac{\pi}{4}=\frac{a+b}{\sqrt{a^2+b^2+c^2} \times \sqrt{2}} \\ & \frac{1}{\sqrt{2}}=\frac{a+b}{\sqrt{a^2+b^2+c^2} \times \sqrt{2}} \\ & \Rightarrow(a+b)^2=a^2+b^2+c^2 \Rightarrow 2 a b=c^2 \end{aligned} $$ which satisfied by the option (b) $$ \therefore a=1, b=1, c=\sqrt{2} $$

Hence, the correct answer is option: 'B'.

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