JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Three Dimensional Geometry

More JEE Main questions on Three Dimensional Geometry

1. Let $(\alpha ,\beta ,\gamma )$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5,4,2)$ on the line $\vec{r}=(-\hat{i}+3\hat{j}+\hat{k})+\lambda (2\hat{i}+3\hat{j}-\hat{k})$.
   Then the length of the projection of the vector …
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2. Let the line L pass through the point $(-3,5,2)$ and make equal angles with the positive coordinate axes. If the distance of L from the point $(-2,r,1)$ is $\sqrt{\frac{14}{3}}$, then the sum of all possible values of $r$ is :
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3. Let the line $L_1$ be parallel to the vector $-3\hat{i}+2\hat{j}+4\hat{k}$ and pass through the point $(2,6,7)$ and the line $L_2$ be parallel to the vector $2\hat{i}+\hat{j}+3\hat{k}$ and pass through the point $(4,3,5)$. If the line $L_3$ is parallel to the vector …
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4. Let $\mathrm{P}(\alpha ,\beta ,\gamma )$ be the point on the line
   $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{-3}=\mathrm{z}$ at a distance $4\sqrt{14}$ from the point $(1,-1,0)$ and nearer to the origin. Then the shortest distance, between the lines …
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5. If the image of the point $\mathrm{P}(1,2,a)$ in the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{7-\mathrm{z}}{2}$ is $\mathrm{Q}(5,\mathrm{b},\mathrm{c})$, then ${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2$ is equal to
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6. Let L be the line $\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6}$ and let S be the set of all points $(a,b,c)$ on L, whose distance from the line $\frac{x+1}{2}=\frac{y+1}{3}=\frac{z-9}{0}$ along the line L is 7. Then ${\sum }_{(a,b,c)\in S}(a+b+c)$ is equal to :
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