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Yukti Vault

Vectors & 3D

3D Geometry: JEE Main & Advanced PYQs

461 previous year questions on 3D Geometry from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 51-100 of 461

  1. Let the line passing through the points $P(2,-1,2)$ and $Q(5,3,4)$ meet the plane $x-y+z=4$ at the point $R$. Then the distance of the point $R$ from the plane …
  2. Let the line $\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}$ intersect the lines $\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}$ and $\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}$ at …
  3. One vertex of a rectangular parallelopiped is at the origin $O$ and the lengths of its edges along $x$, $y$ and $z$ axes are $3,4$ and $5$ units respectively. Let $P$ be …
  4. If the foot of the perpendicular from point $(4,3,8)$ on the line $L_1:\frac{x-a}{l}=\frac{y-2}{3}=\frac{z-b}{4}$, $l\neq0$ is $(3,5,7)$, then the shortest distance …
  5. The shortest distance between the lines $\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$ and $\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$ is
  6. The vector equation of the plane passing through the intersection of the planes $\vec{r}\cdot\left(\hat{i}+\hat{j}+\hat{k}\right)=1$ and …
  7. Let $P$ be the plane passing through the line $\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}$ and the point $(2,4,-3)$. If the image of the point $(-1,3,4)$ in the plane …
  8. A plane passes through the points $A(1,2,3)$, $B(2,3,1)$ and $C(2,4,2)$. If $O$ is the origin and $P$ is $(2,-1,1)$, then the projection of $\overrightarrow{OP}$ on this …
  9. Let $(\alpha,\beta,\gamma)$ be the image of the point $P(2,3,5)$ in the plane $2x+y-3z=6$. Then $\alpha+\beta+\gamma$ is equal to
  10. If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda}$ and $\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{3}}$ …
  11. The plane, passing through the points $(0,-1,2)$ and $(-1,2,1)$ and parallel to the line passing through $(5,1,-7)$ and $(1,-1,-1)$, also passes through the point
  12. Let the equation of plane passing through the line of intersection of the planes $x+2y+az=2$ and $x-y+z=3$ be $5x-11y+bz=6a-1$. For $c\in\mathbb{Z}$, if the distance of …
  13. Let $(\alpha,\beta,\gamma)$ be the mirror image of the point $(2,3,5)$ in the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$. Then, $2\alpha+3\beta+4\gamma$ is equal to
  14. If the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x}{1}=\frac{y}{\alpha}=\frac{z-5}{1}$ is $\frac{5}{\sqrt{6}}$, then the …
  15. If the shortest distance between the lines $\frac{x-\lambda}{2}=\frac{y-4}{3}=\frac{z-3}{4}$ and $\frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8}$ is $\frac{13}{\sqrt{29}}$ …
  16. If the image of the point $P(1,0,3)$ in the line joining the points $A(4,7,1)$ and $B(3,5,3)$ is $Q(\alpha,\beta,\gamma)$, then $\alpha+\beta+\gamma$ is equal to :
  17. Let the values of $\lambda$ for which the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and …
  18. If the shortest distance between the lines $\frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1}$ and $\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1}$ is $1$, then the …
  19. Let the vertices $Q$ and $R$ of the triangle $PQR$ lie on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$, $QR=5$ and the coordinates of the point $P$ be $(0,2,3)$. …
  20. Let the line $L$ intersect the lines $x-2=-y=z-1$, $2(x+1)=2(y-1)=z+1$ and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$. Then which of the …
  21. Let $L_1:\vec{r}=\left(\hat{i}-\hat{j}+2\hat{k}\right)+\lambda\left(\hat{i}-\hat{j}+2\hat{k}\right)$, $\lambda\in\mathbb{R}$ …
  22. Let $PQR$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the mid point of $PQ$. The distance of the centroid of $\triangle PQR$ from the point of intersection of …
  23. The shortest distance, between lines $L_1$ and $L_2$, where $L_1:\frac{x-1}{2}=\frac{y+1}{-3}=\frac{z+4}{2}$ and $L_2$ is the line, passing through the points …
  24. The shortest distance between the lines $\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}$ and $\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}$ is :
  25. The line $L_1$ is parallel to the vector $\vec{a}=-3\hat{i}+2\hat{j}+4\hat{k}$ and passes through the point $(7,6,2)$ and the line $L_2$ is parallel to the vector …
  26. Line $L_1$ passes through the point $(1,2,3)$ and is parallel to $z$-axis. Line $L_2$ passes through the point $(\lambda,5,6)$ and is parallel to $y$-axis. Let for …
  27. Let the foot of perpendicular of the point $P(3,-2,-9)$ on the plane passing through the points $(-1,-2,-3)$, $(9,3,4)$, $(9,-2,1)$ be $Q(\alpha,\beta,\gamma)$. Then the …
  28. Let the points on the plane $P$ be equidistant from the points $(-4,2,1)$ and $(2,-2,3)$. Then the acute angle between the plane $P$ and the plane $2x+y+3z=1$ is
  29. Let a line passing through the point $(4,1,0)$ intersect the line $L_1:\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ at the point $A(\alpha,\beta,\gamma)$ and the line …
  30. Let $P(x,y,z)$ be a point in the first octant, whose projection in the $xy$-plane is the point $Q$. Let $OP=\gamma$; the angle between $OQ$ and the positive $x$-axis be …
  31. The line, that is coplanar to the line $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$, is
  32. Let $Q$ be the mirror image of the point $P(1,0,1)$ with respect to the plane $S:x+y+z=5$. If a line $L$ passing through $(1,-1,-1)$, parallel to the line $PQ$ meets the …
  33. Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$- and $z$-axes, respectively, is half of the angle that this line makes with the …
  34. The distance of the point $Q(0,2,-2)$ form the line passing through the point $P(5,-4,3)$ and perpendicular to the lines …
  35. If the two lines $l_1:\frac{x-2}{3}=\frac{y+1}{-2}$, $z=2$ and $l_2:\frac{x-1}{1}=\frac{2y+3}{\alpha}=\frac{z+5}{2}$ are perpendicular, then an angle between the lines …
  36. Let the plane $2x+3y+z+20=0$ be rotated through a right angle about its line of intersection with the plane $x-3y+5z=8$. If the mirror image of the point …
  37. The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is
  38. The distance of the point $(7,10,11)$ from the line $\frac{x-4}{1}=\frac{y-4}{0}=\frac{z-2}{3}$ along the line $\frac{x-9}{2}=\frac{y-13}{3}=\frac{z-17}{6}$ is
  39. Let $P(\alpha,\beta,\gamma)$ be the image of the point $Q(3,-3,1)$ in the line $\frac{x-0}{1}=\frac{y-3}{1}=\frac{z-1}{-1}$ and $R$ be the point $(2,5,-1)$. If the area …
  40. Let $(\alpha,\beta,\gamma)$ be the foot of perpendicular from the point $(1,2,3)$ on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$. Then $19(\alpha+\beta+\gamma)$ …
  41. Consider the lines $L_1:x-1=y-2=z$ and $L_2:x-2=y=z-1$. Let the feet of the perpendiculars from the point $P(5,1,-3)$ on the lines $L_1$ and $L_2$ be $Q$ and $R$ …
  42. If the plane $2x+y-5z=0$ is rotated about its line of intersection with the plane $3x-y+4z-7=0$ by an angle of $\frac{\pi}{2}$, then the plane after the rotation passes …
  43. Let the line $L$ pass through $(1,1,1)$ and intersect the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z}{1}$. Then, which of …
  44. Let $P$ and $Q$ be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of $6$ units from the point $R(1,2,3)$. If the centroid of …
  45. Let the shortest distance between the lines $\frac{x-3}{3}=\frac{y-\alpha}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-\beta}{4}$ be $3\sqrt{30}$. Then …
  46. If $A(3,1,-1)$, $B\left(\frac{5}{3},\frac{7}{3},\frac{1}{3}\right)$, $C(2,2,1)$ and $D\left(\frac{10}{3},\frac{2}{3},\frac{-1}{3}\right)$ are the vertices of a …
  47. Let $A$ be the point of intersection of the lines $L_1:\frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1}$ and $L_2:\frac{x-1}{3}=\frac{y+3}{4}=\frac{z+7}{5}$. Let $B$ and $C$ …
  48. Consider the line $L$ passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\frac{11}{3},\frac{11}{3},\frac{19}{3}\right)$ from the line …
  49. Let $S$ be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{z+6}{1}$ and …
  50. If the equation of the line passing through the point $\left(0,-\frac{1}{2},0\right)$ and perpendicular to the lines …

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