If λ₁ < λ₂ are two values of λ such that the angle between the planes P₁: r(3 i-5 j+k)=7 and P₂: r ·(λ i+j-3 k)=9 is sin ⁻¹((2 √(6))/5), then the square of the length of perpendicular from the point (38 λ₁, 10 λ₂, 2) to the plane P₁ is _ _ _ _ _ _ _.

Three Dimensional Geometry: JEE Main 2023 Mathematics question

JEE Main 2023 · 30 Jan, Morning shift · Q27

JEE Main 202330 Jan, Morning shiftMathematicsHardThree Dimensional Geometry249× since 2002

\lambda_{1} < \lambda_{2}

are two values of

\lambda

such that the angle between the planes

P_{1}: \vec{r}(3 \hat{i}-5 \hat{j}+\hat{k})=7

and

P_{2}: \vec{r} \cdot(\lambda \hat{i}+\hat{j}-3 \hat{k})=9

\sin ^{-1}\left(\frac{2 \sqrt{6}}{5}\right)

, then the square of the length of perpendicular from the point

\left(38 \lambda_{1}, 10 \lambda_{2}, 2\right)

to the plane

P_{1}

is ______________.

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