JEE Mains · Maths · STD 12 - 11. three dimension geometry
Let \(\theta\) be the angle between the planes \(P_1=\vec{r} \cdot(\hat{ i }+\hat{ j }+2 \hat{ k })=9\) and \(P _2=\overrightarrow{ r } \cdot(2 \hat{ i }-\hat{ j }+\hat{ k })=15\).Let \(L\) be the line that meets \(P _2\) at the point \((4,-2,5)\) and makes an angle \(\theta\) with the normal of \(P_{2^*}\) If \(\alpha\) is the angle between \(L\) and \(P_2\) then \(\left(\tan ^2 \theta\right)\left(\cot ^2 \alpha\right)\) is equal to \(...........\).
- A \(9\)
- B \(12\)
- C \(3\)
- D \(63\)
Answer & Solution
Correct Answer
(A) \(9\)
Step-by-step Solution
Detailed explanation
\(\cos \theta=\frac{(\hat{ i }+\hat{ j }+2 \hat{ k }) \cdot(2 \hat{ i }-\hat{ j }+\hat{ k })}{6}=\frac{2-1+2}{6}=\frac{1}{2}\) \(\theta=\pi / 3 \quad \alpha=\pi / 6\) \(\left(\tan ^2 \theta\right)\left(\cot ^2 \alpha\right)\) \((3) (3)=9\)
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