JEE Mathematics / Practice

3D Geometry

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JEE Main 2022 (Online) 30th July Evening ShiftQuestion 201

Let $L_1$ be a line in the $yz$-plane with $y$ and $z$ intercepts $\frac14$ and $\frac1C$ ($C>0$) respectively. Let $L_2$ be the line in the $xz$-plane with $x$ and $z$ intercepts $\frac13$ and $-\frac1C$ respectively. If the shortest distance between $L_1$ and $L_2$ is $\frac15$, then $C^2$ is equal to ________.

3D Geometry

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JEE Main 2021 (Online) 27th July Evening ShiftQuestion 202

The distance of the point $P(3,4,4)$ from the point of intersection of the line joining the points $Q(3,-4,-5)$ and $R(2,-3,1)$ and the plane $2x+y+z=7$, is equal to ________.

3D Geometry

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JEE Main 2023 (Online) 11th April Evening ShiftQuestion 203

Let the line $l:x=\frac{1-y}{-2}=\frac{z-3}{\lambda}$, $\lambda\in\mathbb{R}$ meet the plane $P:x+2y+3z=4$ at the point $(\alpha,\beta,\gamma)$. If the angle between the line $l$ and the plane $P$ is $\cos^{-1}\left(\sqrt{\frac{5}{14}}\right)$, then $\alpha+2\beta+6\gamma$ is equal to $\_\_\_\_\_\_$ .

3D Geometry

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JEE Main 2023 (Online) 13th April Morning ShiftQuestion 204

Let the image of the point $\left(\frac{5}{3},\frac{5}{3},\frac{8}{3}\right)$ in the plane $x-2y+z-2=0$ be $P$. If the distance of the point $Q(6,-2,\alpha)$, $\alpha>0$, from $P$ is $13$, then $\alpha$ is equal to $\_\_\_\_\_\_$ .

3D Geometry

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JEE Main 2022 (Online) 28th July Morning ShiftQuestion 205

Let $P(-2,-1,1)$ and $Q\left(\frac{56}{17},\frac{43}{17},\frac{111}{17}\right)$ be the vertices of the rhombus $PRQS$. If the direction ratios of the diagonal $RS$ are $\alpha,-1,\beta$, where both $\alpha$ and $\beta$ are integers of minimum absolute values, then $\alpha^2+\beta^2$ is equal to ________.

3D Geometry

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