JEE Mathematics / Practice

3D Geometry

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JEE Main 2021 (Online) 20th July Morning ShiftQuestion 346

If the shortest distance between the lines $\vec r_1=\alpha\hat i+2\hat j+2\hat k+\lambda\left(\hat i-2\hat j+2\hat k\right)$, $\lambda\in R$, $\alpha>0$ and $\vec r_2=-4\hat i-\hat k+\mu\left(3\hat i-2\hat j-2\hat k\right)$, $\mu\in R$ is $9$, then $\alpha$ is equal to

3D Geometry

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JEE Main 2023 (Online) 31st January Morning ShiftQuestion 347

Let $\theta$ be the angle between the planes $P_1:\vec r\cdot\left(\hat i+\hat j+2\hat k\right)=9$ and $P_2:\vec r\cdot\left(2\hat i-\hat j+\hat k\right)=15$. 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 $\tan^{2}\theta\cot^{2}\alpha$ is equal to ______.

3D Geometry

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JEE Main 2024 (Online) 27th January Evening ShiftQuestion 348

The lines $\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}$ and $\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}$ intersect at the point $P$. If the distance of $P$ from the line $\frac{x+1}{2}=\frac{y-1}{3}=\frac{z-1}{1}$ is $l$, then $14l^2$ is equal to _____.

3D Geometry

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JEE Main 2024 (Online) 29th January Evening ShiftQuestion 349

Let $O$ be the origin, and $M$ and $N$ be the points on the lines $\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}$ and $\frac{x+8}{12}=\frac{y+2}{5}=\frac{z+11}{9}$ respectively such that $MN$ is the shortest distance between the given lines. Then $\overrightarrow{OM}\cdot\overrightarrow{ON}$ is equal to ______.

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JEE Main 2023 (Online) 30th January Evening ShiftQuestion 350

Let a line $L$ pass through the point $P(2,3,1)$ and be parallel to the line $x+3y-2z-2=0=x-y+2z$. If the distance of $L$ from the point $(5,3,8)$ is $\alpha$, then $3\alpha^{2}$ is equal to

3D Geometry

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