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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 151-200 of 461

  1. Let $P$ be an arbitrary point having sum of the squares of the distances from the planes $x+y+z=0$, $lx-nz=0$ and $x-2y+z=0$, equal to $9$. If the locus of the point $P$ …
  2. The mirror image of the point $(1,2,3)$ in a plane is $\left(-\frac73,-\frac43,-\frac13\right)$. Which of the following points lies on this plane ?
  3. If the distance of the point $(1,-2,3)$ from the plane $x+2y-3z+10=0$ measured parallel to the line, $\frac{x-1}{3}=\frac{2-y}{m}=\frac{z+3}{1}$ is $\sqrt{\frac{7}{2}}$ …
  4. Let a line $L_1$ pass through the origin and be perpendicular to the lines $L_2:\vec{r}=(3+t)\hat{i}+(2t-1)\hat{j}+(2t+4)\hat{k}$ and …
  5. A line '$l$' passing through origin is perpendicular to the lines $l_1:\vec{r}=(3+t)\hat{i}+(-1+2t)\hat{j}+(4+2t)\hat{k}$ …
  6. Let $P$ be a plane containing the line $\frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2}$ and parallel to the line $\frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7}$. If the point …
  7. Let the image of the point $P(0,-5,0)$ in the line $\frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2}$ be the point $R$ and the image of the point …
  8. Let $\lambda$ be an integer. If the shortest distance between the lines $x-\lambda=2y-1=-2z$ and $x=y+2\lambda=z-\lambda$ is $\frac{\sqrt{7}}{2\sqrt{2}}$, then the value …
  9. The square of the distance of the point of intersection of the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}$ and the plane $2x-y+z=6$ from the point $(-1,-1,2)$ is …
  10. Consider a triangle $ABC$ whose vertices are $A(0,\alpha,\alpha)$, $B(\alpha,0,\alpha)$ and $C(\alpha,\alpha,0)$, $\alpha>0$. Let $D$ be a point moving on the line …
  11. Let $P(\alpha,\beta,\gamma)$ be the image of the point $Q(1,6,4)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. Then $2\alpha+\beta+\gamma$ is equal to …
  12. Let the area of the triangle formed by the lines $x+2=y-1=z$ and $\frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1}$ and $\frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1}$ be $A$. Then …
  13. Maximum number of lines, through the origin which make equal angles with the three coordinate axes in $XYZ$-space, is $\_\_\_\_\_\_$ .
  14. Let the line of the shortest distance between the lines $L_1:\vec{r}=\left(\hat{i}+2\hat{j}+3\hat{k}\right)+\lambda\left(\hat{i}-\hat{j}+\hat{k}\right)$ and …
  15. Let $Q$ and $R$ be two points on the line $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}$ at a distance $\sqrt{26}$ from the point $P(4,2,7)$. Then the square of the area of …
  16. Let a line $l$ pass through the origin and be perpendicular to the lines …
  17. If the lines $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}$ intersect, then the magnitude of the minimum value of …
  18. Let $\lambda_1,\lambda_2$ be the values of $\lambda$ for which the points $\left(\frac{5}{2},1,\lambda\right)$ and $(-2,0,1)$ are at equal distance from the plane …
  19. If the shortest distance between the lines $\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}$ is $\frac{44}{\sqrt{30}}$ …
  20. Let a line passing through the point $(-1,2,3)$ intersect the lines $L_1:\frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$ at $M(\alpha,\beta,\gamma)$ and …
  21. Let $d$ be the distance between the foot of perpendiculars of the points $P(1,2,-1)$ and $Q(2,-1,3)$ on the plane $-x+y+z=1$. Then $d^{2}$ is equal to $\_\_\_\_\_\_$ .
  22. Let $P$ be the point $(10,-2,-1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1,7,6)$ on the line passing through the points $(2,-5,11)$ and …
  23. Let $Q$ be the foot of the perpendicular from the point $P(7,-2,13)$ on the plane containing the lines $\frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}$ and …
  24. If the distance between the plane, $23x-10y-2z+48=0$ and the plane containing the lines $\frac{x+1}2=\frac{y-3}4=\frac{z+1}3$ and …
  25. A line with direction ratios $2,1,2$ meets the lines $x=y+2=z$ and $x+2=2y=2z$ respectively at the points $P$ and $Q$. If the length of the perpendicular from the point …
  26. Let the image of the point $P(1,2,3)$ in the line $L:\frac{x-6}{3}=\frac{y-1}{2}=\frac{z-2}{3}$ be $Q$. Let $R(\alpha,\beta,\gamma)$ be a point that divides internally …
  27. Let $P_1$ be the plane $3x-y-7z=11$ and $P_2$ be the plane passing through the points $(2,-1,0)$, $(2,0,-1)$, and $(5,1,1)$. If the foot of the perpendicular drawn from …
  28. Let the image of the point $P(1,2,3)$ in the plane $2x-y+z=9$ be $Q$. If the coordinates of the point $R$ are $(6,10,7)$, then the square of the area of the triangle …
  29. Let $P_1:\vec{r}\cdot\left(2\hat{i}+\hat{j}-3\hat{k}\right)=4$ be a plane. Let $P_2$ be another plane which passes through the points $(2,-3,2)$, $(2,-2,-3)$ and …
  30. Let a line having direction ratios $1,-4,2$ intersect the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the points …
  31. Let the plane $x+3y-2z+6=0$ meet the co-ordinate axes at the points $A,B,C$. If the orthocenter of the triangle $ABC$ is $\left(\alpha,\beta,\frac{6}{7}\right)$, then …
  32. The largest value of $a$, for which the perpendicular distance of the plane containing the lines $\vec r=(\hat i+\hat j)+\lambda(\hat i+a\hat j-\hat k)$ and …
  33. Let the lines $L_1:\vec{r}=\lambda\left(\hat{i}+2\hat{j}+3\hat{k}\right)$, $\lambda\in\mathbf{R}$ …
  34. Let the plane $P$ contain the line $2x+y-z-3=0=5x-3y+4z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then the distance of the point …
  35. A line passes through $A(4,-6,-2)$ and $B(16,-2,4)$. The point $P(a,b,c)$, where $a,b,c$ are non-negative integers, on the line $AB$ lies at a distance of $21$ units …
  36. Let a plane $P$ pass through the point $(3,7,-7)$ and contain the line $\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$. If distance of the plane $P$ from the origin is $d$ …
  37. The plane passing through the line $L:lx-y+3(1-l)z=1$, $x+2y-z=2$ and perpendicular to the plane $3x+2y+z=6$ is $3x-8y+7z=4$. If $\theta$ is the acute angle between the …
  38. Suppose the line $\frac{x-2}{\alpha}=\frac{y-2}{-5}=\frac{z+2}{2}$ lies on the plane $x+3y-2z+\beta=0$. Then $(\alpha+\beta)$ is equal to $\_\_\_\_\_\_$ .
  39. Let the line $L$ be the projection of the line $\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}$ in the plane $x-2y-z=3$. If $d$ is the distance of the point $(0,0,6)$ from …
  40. Let a line with direction ratios $a,-4a,-7$ be perpendicular to the lines with direction ratios $3,-1,2b$ and $b,a,-2$. If the point of intersection of the line …
  41. If the shortest distance between the lines $\vec{r}=\left(-\hat{i}+3\hat{k}\right)+\lambda\left(\hat{i}-a\hat{j}\right)$ and …
  42. Let $Q$ and $R$ be the feet of perpendiculars from the point $P(a,a,a)$ on the lines $x=y,\ z=1$ and $x=-y,\ z=-1$ respectively. If $\angle QPR$ is a right angle, then …
  43. If the lines $\frac{x-k}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x+1}{3}=\frac{y+2}{2}=\frac{z+3}{1}$ are co-planar, then the value of $k$ is $\_\_\_\_\_\_$ .
  44. The square of the distance of the image of the point $(6,1,5)$ in the line $\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}$, from the origin is $\_\_\_\_\_\_$ .
  45. Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2x-y+z+3=0$ and let $R(3,5,\gamma)$ be a point of this plane. Then the square of the …
  46. Let the foot of perpendicular from the point $A(4,3,1)$ on the plane $P:x-y+2z+3=0$ be $N$. If $B(5,\alpha,\beta)$, $\alpha,\beta\in\mathbb{Z}$ is a point on plane $P$ …
  47. Let the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}$ intersect the plane containing the lines $\frac{x-4}{1}=\frac{y+1}{-2}=\frac z1$ and …
  48. Let a plane $P$ contain two lines $\vec r=\hat i+\lambda(\hat i+\hat j)$, $\lambda\in\mathbf{R}$ and $\vec r=-\hat j+\mu(\hat j-\hat k)$, $\mu\in\mathbf{R}$. If …
  49. Let $l_1$ be the line in $xy$-plane with $x$ and $y$ intercepts $\frac{1}{8}$ and $\frac{1}{4\sqrt{2}}$ respectively, and $l_2$ be the line in $zx$-plane with $x$ and …
  50. Let the mirror image of the point $(a,b,c)$ with respect to the plane $3x-4y+12z+19=0$ be $(a-6,\beta,\gamma)$. If $a+b+c=5$, then $7\beta-9\gamma$ is equal to …

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