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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 101-150 of 461

  1. Let $A$ and $B$ be two distinct points on the line $L:\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Both $A$ and $B$ are at a distance $2\sqrt{17}$ from the foot of …
  2. If the shortest distance between the lines $\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}$ and $\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}$ is $\frac{6}{\sqrt{5}}$ …
  3. Let $P$ be the plane passing through the intersection of the planes $\vec{r}\cdot\left(\hat{i}+3\hat{j}-\hat{k}\right)=5$ and …
  4. If the shortest distance between the lines $L_1:\vec{r}=(2+\lambda)\hat{i}+(1-3\lambda)\hat{j}+(3+4\lambda)\hat{k}$, $\lambda\in\mathbb{R}$ …
  5. If the mirror image of the point $P(3,4,9)$ in the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}$ is $(\alpha,\beta,\gamma)$, then $14(\alpha+\beta+\gamma)$ is :
  6. The distance, of the point $(7,-2,11)$ from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is :
  7. Let the values of $p$, for which the shortest distance between the lines $\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and …
  8. The distance of the point $(-1,2,3)$ from the plane $\vec{r}\cdot\left(\hat{i}-2\hat{j}+3\hat{k}\right)=10$ parallel to the line of the shortest distance between the …
  9. Let $N$ be the foot of perpendicular from the point $P(1,-2,3)$ on the line passing through the points $(4,5,8)$ and $(1,-7,5)$. Then the distance of $N$ from the plane …
  10. The length of the perpendicular from the point $(1,-2,5)$ on the line passing through $(1,2,4)$ and parallel to the line $x+y-z=0=x-2y+3z-5$ is :
  11. A plane passing through the point $(3,1,1)$ contains two lines whose direction ratios are $1,-2,2$ and $2,3,-1$ respectively. If this plane also passes through the point …
  12. Let $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z+3}{-1}$ lie on the plane $px-qy+z=5$, for some $p,q\in\mathbf{R}$. The shortest distance of the plane from the origin is:
  13. Let the lines $\frac{x-1}{\lambda}=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{x+26}{-2}=\frac{y+18}{3}=\frac{z+28}{\lambda}$ be coplanar and $P$ be the plane containing …
  14. The foot of the perpendicular drawn from the point $(4,2,3)$ to the line joining the points $(1,-2,3)$ and $(1,1,0)$ lies on the plane :
  15. If the equation of the plane passing through the line of intersection of the planes $2x-7y+4z-3=0$, $3x-5y+4z+11=0$ and the point $(-2,1,3)$ is $ax+by+cz-7=0$, then the …
  16. If the lines $\vec{r}=\left(\hat{i}-\hat{j}+\hat{k}\right)+\lambda\left(3\hat{j}-\hat{k}\right)$ and …
  17. Let the plane $P:\vec{r}\cdot\vec{a}=d$ contain the line of intersection of two planes $\vec{r}\cdot\left(\hat{i}+3\hat{j}-\hat{k}\right)=6$ and …
  18. The shortest distance between the lines $\frac{x+7}{-6}=\frac{y-6}{7}=z$ and $\frac{7-x}{2}=y-2=z-6$ is
  19. A plane $P$ is parallel to two lines whose direction ratios are $-2,1,-3$ and $-1,2,-2$ and it contains the point $(2,2,-2)$. Let $P$ intersect the co-ordinate axes at …
  20. Let $Q$ be the mirror image of the point $P(1,2,1)$ with respect to the plane $x+2y+2z=16$. Let $T$ be a plane passing through the point $Q$ and contains the line …
  21. A plane $P$ meets the coordinate axes at $A$, $B$ and $C$ respectively. The centroid of $\Delta ABC$ is given to be $(1,1,2)$. Then the equation of the line through this …
  22. The shortest distance between the lines $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}$ and $\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}$, is
  23. If two straight lines whose direction cosines are given by the relations $l+m-n=0$, $3l^{2}+m^{2}+cnl=0$ are parallel, then the positive value of $c$ is :
  24. Let $Q$ be the foot of perpendicular drawn from the point $P(1,2,3)$ to the plane $x+2y+z=14$. If $R$ is a point on the plane such that $\angle PRQ=60^\circ$, then the …
  25. If the length of the perpendicular drawn from the point $P(a,4,2)$, $a>0$ on the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1}$ is $2\sqrt6$ units and …
  26. Let the foot of the perpendicular from the point $(1,2,4)$ on the line $\frac{x+2}{4}=\frac{y-1}{2}=\frac{z+1}{3}$ be $P$. Then the distance of $P$ from the plane …
  27. The distance of the point $(3,2,-1)$ from the plane $3x-y+4z+1=0$ along the line $\frac{2-x}{2}=\frac{y-3}{2}=\frac{z+1}{1}$ is equal to :
  28. A plane $E$ is perpendicular to the two planes $2x-2y+z=0$ and $x-y+2z=4$, and passes through the point $P(1,-1,1)$. If the distance of the plane $E$ from the point …
  29. The plane passing through the points $(1,2,1)$, $(2,1,2)$ and parallel to the line, $2x=3y$, $z=1$ also passes through the point :
  30. The lines $\vec r=(\hat i-\hat j)+l(2\hat i+\hat k)$ and $\vec r=(2\hat i-\hat j)+m(\hat i+\hat j-\hat k)$
  31. If the plane $P$ passes through the intersection of two mutually perpendicular planes $2x+ky-5z=1$ and $3kx-ky+z=5$, $k<3$ and intercepts a unit length on positive …
  32. The acute angle between the planes $P_1$ and $P_2$, when $P_1$ and $P_2$ are the planes passing through the intersection of the planes $5x+8y+13z-29=0$ and …
  33. Let the plane $ax+by+cz=d$ pass through $(2,3,-5)$ and is perpendicular to the planes $2x+y-5z=10$ and $3x+5y-7z=12$. If $a,b,c,d$ are integers $d>0$ and …
  34. If two distinct points $Q,R$ lie on the line of intersection of the planes $-x+2y-z=0$ and $3x-5y+2z=0$ and $PQ=PR=\sqrt{18}$ where the point $P$ is $(1,-2,3)$, then the …
  35. If $(a,b,c)$ is the image of the point $(1,2,-3)$ in the line, $\frac{x+1}2=\frac{y-3}{-2}=\frac z{-1}$, then $a+b+c$ is equal to :
  36. If the mirror image of the point $(2,4,7)$ in the plane $3x-y+4z=2$ is $(a,b,c)$, then $2a+b+2c$ is equal to :
  37. The plane which bisects the line joining the points $(4,-2,3)$ and $(2,4,-1)$ at right angles also passes through the point :
  38. If $(2,3,9)$, $(5,2,1)$, $(1,\lambda,8)$ and $(\lambda,2,3)$ are coplanar, then the product of all possible values of $\lambda$ is :
  39. The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to the line $\frac x2=\frac y3=\frac z{-6}$ is :
  40. Let $ax+by+cz+3=0$ be the plane which bisects the acute angle between the planes $2x-y-2z+3=0$ and $3x-2y+6z+8=0$. Then $a+b+c$ is equal to
  41. Let the plane $ax+by+cz+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at the right angles. If $a,b,c,d$ are integers, then the minimum value of …
  42. The foot of the perpendicular from a point on the circle $x^2+y^2=1$, $z=0$ to the plane $2x+3y+z=6$ lies on which one of the following curves ?
  43. If the foot of the perpendicular from the point $A(-1,4,3)$ on the plane $P:2x+my+nz=4$, is $\left(-2,\frac72,\frac32\right)$, then the distance of the point $A$ from …
  44. The shortest distance between the lines $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}$ is :
  45. Let the mirror image of the point $(1,3,a)$ with respect to the plane $\vec{r}\cdot\left(2\hat{i}-\hat{j}+\hat{k}\right)-b=0$ be $(-3,5,2)$. Then, the value of $|a+b|$ …
  46. For real numbers $\alpha$ and $\beta\ne0$, if the point of intersection of the straight lines $\frac{x-\alpha}{1}=\frac{y-1}{2}=\frac{z-1}{3}$ and …
  47. The equation of the planes parallel to the plane $x-2y+2z-3=0$ which are at unit distance from the point $(1,2,3)$ is $ax+by+cz+d=0$. If $(b-d)=K(c-a)$, then the …
  48. Let $(\lambda,2,1)$ be a point on the plane which passes through the point $(4,-2,2)$. If the plane is perpendicular to the line joining the points $(-2,-21,29)$ and …
  49. Let $P$ be the plane containing the straight line $\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$ and perpendicular to the plane containing the straight lines …
  50. Let the plane passing through the point $(-1,0,-2)$ and perpendicular to each of the planes $2x+y-z=2$ and $x-y-z=3$ be $ax+by+cz+8=0$. Then the value of $a+b+c$ is …

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