Skip to main content
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.

Practice this chapter

Questions 401-450 of 461

  1. Let $P$ be a point in the first octant, whose image $Q$ in the plane $x+y=3$ (that is, the line segment $PQ$ is perpendicular to the plane $x+y=3$ and the mid-point of …
  2. Let $\alpha,\beta,\gamma,\delta$ be real numbers such that $\alpha^2+\beta^2+\gamma^2\neq0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is the mirror image of …
  3. Let $L_1$ and $L_2$ be the following straight lines. $$L_1:\frac{x-1}{1}=\frac{y}{-1}=\frac{z-1}{3}\ \text{ and }\ L_2:\frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}.$$ …
  4. If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{m}{n}$ …
  5. If the image of the point $(4,4,3)$ in the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-1}{3}$ is $(\alpha,\beta,\gamma)$, then $\alpha+\beta+\gamma$ is equal to
  6. Let $A(x,y,z)$ be a point in $xy$-plane, which is equidistant from three points $(0,3,2)$, $(2,0,3)$ and $(0,0,1)$. Let $B=(1,4,-1)$ and $C=(2,0,-2)$. Then among the …
  7. The distance of the line $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$ from the point $(1,4,0)$ along the line $\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}$ is :
  8. The unit vector perpendicular to both $L_1$ and $L_2$ is
  9. The value of $D$ is ___ .
  10. The shortest distance between $L_1$ and $L_2$ is
  11. The distance of the point $(1,1,1)$ from the plane passing through the point $(-1,-2,-1)$ and whose normal is perpendicular to both the lines $L_1$ and $L_2$ is
  12. An equation of a plane parallel to the plane $x-2y+2z-5=0$ and at a unit distance from the origin is :
  13. If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac z1$ intersect, then $k$ is equal to :
  14. $P_1$ and $P_2$ are planes passing through the origin. $L_1$ and $L_2$ are two lines on $P_1$ and $P_2$ respectively such that their intersection is the origin. Show …
  15. The equation of a line passing through the point $(2,-1,1)$ and the point of intersection of the lines $2x-y-4=0=y+2z$ and $x+3z=0=2x+5z-1$, is
  16. The distance of the point $(5,6,9)$ from the plane, which passes through the line of intersection of the planes $x+y-z=1$ and $2x+3y+4z=5$, and is perpendicular to the …
  17. Let $P$ be a plane $lx+my+nz=0$ containing the line, $\dfrac{1-x}{1}=\dfrac{y+4}{2}=\dfrac{z+2}{3}$. If plane $P$ divides the line segment $AB$ joining points …
  18. If for $a>0$, the feet of perpendiculars from the points $A(a,-2a,3)$ and $B(0,4,5)$ on the plane $lx+my+nz=0$ are points $C(0,-a,-1)$ and $D$ respectively, then the …
  19. Let the position vectors of two points $P$ and $Q$ be $3\hat{i}-\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}-4\hat{k}$, respectively. Let $R$ and $S$ be two points such that …
  20. Let the point, on the line passing through the points $P(1,-2,3)$ and $Q(5,-4,7)$, farther from the origin and at a distance of $9$ units from the point $P$, be …
  21. If the shortest distance between the lines $\dfrac{x+2}{2}=\dfrac{y+3}{3}=\dfrac{z-5}{4}$ and $\dfrac{x-3}{1}=\dfrac{y-2}{-3}=\dfrac{z+4}{2}$ is …
  22. Let $P$ be the point of intersection of the lines $\dfrac{x-2}{1}=\dfrac{y-4}{5}=\dfrac{z-2}{1}$ and $\dfrac{x-3}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{2}$. Then, the shortest …
  23. Consider a line $L$ passing through the points $P(1,2,1)$ and $Q(2,1,-1)$. If the mirror image of the point $A(2,2,2)$ in the line $L$ is $(\alpha,\beta,\gamma)$, then …
  24. Let $d$ be the distance of the point of intersection of the lines $\dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1}$ and $\dfrac{x-7}{4}=\dfrac{y-9}{3}=\dfrac{z-4}{2}$ from …
  25. If the line $\dfrac{2-x}{3}=\dfrac{3y-2}{4\lambda+1}=4-z$ makes a right angle with the line $\dfrac{x+3}{3\mu}=\dfrac{1-2y}{6}=\dfrac{5-z}{7}$, then $4\lambda+9\mu$ is …
  26. Let the point $(-1,\alpha,\beta)$ lie on the line of the shortest distance between the lines $\dfrac{x+2}{-3}=\dfrac{y-2}{4}=\dfrac{z-5}{2}$ and …
  27. Let $(\alpha,\beta,\gamma)$ be the image of the point $(8,5,7)$ in the line $\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{5}$. Then $\alpha+\beta+\gamma$ is equal to :
  28. If for some $\alpha\in\mathbf{R}$, the lines $L_1:\dfrac{x+1}{2}=\dfrac{y-2}{-1}=\dfrac{z-1}{1}$ and $L_2:\dfrac{x+2}{\alpha}=\dfrac{y+1}{5-\alpha}=\dfrac{z+1}{1}$ are …
  29. The shortest distance between the lines $\dfrac{x-1}{0}=\dfrac{y+1}{-1}=\dfrac{z}{1}$ and $x+y+z+1=0$, $2x-y+z+3=0$ is :
  30. Let $L_1:\dfrac{x-1}{3}=\dfrac{y-1}{-1}=\dfrac{z+1}{0}$ and $L_2:\dfrac{x-2}{2}=\dfrac{y}{0}=\dfrac{z+4}{\alpha}$, $\alpha\in\mathbf{R}$, be two lines, which intersect …
  31. Let $L_1:\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $L_2:\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}$ be two lines. Then which of the following points lies on …
  32. The perpendicular distance, of the line $\dfrac{x-1}{2}=\dfrac{y+2}{-1}=\dfrac{z+3}{2}$ from the point $P(2,-10,1)$, is :
  33. Let a line pass through two distinct points $P(-2,-1,3)$ and $Q$, and be parallel to the vector $3\hat{i}+2\hat{j}+2\hat{k}$. If the distance of the point $Q$ from the …
  34. Let the position vectors of the vertices $A$, $B$ and $C$ of a tetrahedron $ABCD$ be $\hat{i}+2\hat{j}+\hat{k}$, $\hat{i}+3\hat{j}-2\hat{k}$ and …
  35. Let $P$ be the foot of the perpendicular from the point $Q(10,-3,-1)$ on the line $\dfrac{x-3}{7}=\dfrac{y-2}{-1}=\dfrac{z+1}{-2}$. Then the area of the right angled …
  36. Let $P$ be the image of the point $Q(7,-2,5)$ in the line $L:\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z}{4}$ and $R(5,p,q)$ be a point on $L$. Then the square of the area of …
  37. The square of the distance of the point $\left(\dfrac{15}{7},\dfrac{32}{7},7\right)$ from the line $\dfrac{x+1}{3}=\dfrac{y+3}{5}=\dfrac{z+5}{7}$ in the direction of the …
  38. The direction ratios of the normal to the plane through $(1,0,0)$, $(0,1,0)$ which makes an angle $\pi/4$ with the plane $x+y=3$ are :
  39. A plane which passes through the point $(3,2,0)$ and the line $\frac{x-4}{1}=\frac{y-7}{5}=\frac{z-4}{4}$ is :
  40. Two systems of rectangular axes have the same origin. If a plane cuts them at distances $a,b,c$ and $a',b',c'$ from the origin, then
  41. The radius of the circle in which the sphere $x^{2}+y^{2}+z^{2}+2x-2y-4z-19=0$ is cut by the plane $x+2y+2z+7=0$ is
  42. The shortest distance from the plane $12x+4y+3z=327$ to the sphere $x^{2}+y^{2}+z^{2}+4x-2y-6z=155$ is
  43. Distance between two parallel planes $2x+y+2z=8$ and $4x+2y+4z+5=0$ is :
  44. The intersection of the spheres $x^{2}+y^{2}+z^{2}+7x-2y-z=13$ and $x^{2}+y^{2}+z^{2}-3x+3y+4z=8$ is the same as the intersection of one of the spheres and the plane
  45. If the angle $\theta$ between the line $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2x-y+\sqrt\lambda\,z+4=0$ is such that $\sin\theta=\frac13$, then the …
  46. If the plane $2ax-3ay+4az+6=0$ passes through the midpoint of the line joining the centres of the spheres $x^{2}+y^{2}+z^{2}+6x-8y-2z=13$ and …
  47. The distance between the line $\vec r=2\hat i-2\hat j+3\hat k+\lambda\left(\hat i-\hat j+4\hat k\right)$ and the plane $\vec r\cdot\left(\hat i+5\hat j+\hat k\right)=5$ …
  48. The plane $x+2y-z=4$ cuts the sphere $x^{2}+y^{2}+z^{2}-x+z-2=0$ in a circle of radius
  49. The image of the point $(-1,3,4)$ in the plane $x-2y=0$ is :
  50. If $(2,3,5)$ is one end of a diameter of the sphere $x^{2}+y^{2}+z^{2}-6x-12y-2z+20=0$, then the coordinates of the other end of the diameter are

Other chapters