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INTRODUCTION TO THREE DIMENSIONAL GEOMETRY-Objective Questions for Entrance Exams

The chapter Introduction to Three Dimensional Geometry forms the conceptual gateway from planar coordinate geometry to spatial reasoning, a transition that is critically tested in competitive examinations. The objective questions presented here are carefully curated to reflect the actual and repeatedly modelled patterns seen in JEE (Main & Advanced), NEET, AIIMS, BITSAT, KVPY, Olympiads, and various state engineering entrance examinations over the years. These MCQs emphasize foundational ideas such as coordinate axes, octants, distance formula, section formula, symmetry, locus of points, and distances from axes and planes—topics that are deceptively simple yet frequently decisive in exams. Each question is framed to test clarity of concepts rather than mechanical computation, aligning closely with NCERT while extending to competitive-level rigor. Concise explanations accompany every answer to reinforce geometric intuition and eliminate common misconceptions. Practicing these questions will help aspirants build speed, accuracy, and confidence in handling three-dimensional coordinate geometry under exam conditions.

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Exercise

INTRODUCTION TO THREE DIMENSIONAL GEOMETRY

by Academia Aeternum

1. If the coordinates of a point \(P\) are \((2, -3, 5)\), then the distance of \(P\) from the \(xy\)-plane is
(Exam: IIT-JEE Year: 1998)
2. The coordinates of a point on the \(x\)-axis at a distance \(7\) from the origin can be
(Exam: AIEEE Year: 2004)
3. The distance between the points \(A(1, 2, 3)\) and \(B(4, 6, 3)\) is
(Exam: NEET Year: 2016)
4. The coordinates of the foot of the perpendicular from the origin to the \(yz\)-plane are
(Exam: IIT-JEE Year: 2001)
5. A point \(P(x, y, z)\) lies on the \(xz\)-plane if
(Exam: BITSAT Year: 2012)
6. The distance of the point \((3, -4, 12)\) from the origin is
(Exam: KVPY Year: 2014)
7. The locus of a point equidistant from the \(xy\)-plane and \(yz\)-plane is
(Exam: IIT-JEE Year: 2005)
8. The point which is equidistant from all the three coordinate axes is
(Exam: AIIMS Year: 2010)
9. The coordinates of the midpoint of the line joining \((2, -4, 6)\) and \((-2, 4, 10)\) are
(Exam: JEE Main Year: 2019)
10. The coordinates of a point on the \(y\)-axis at a distance \(5\) from the origin are
(Exam: State CET Year: 2015)
11. If \(A(1, 1, 1)\), \(B(2, 3, 4)\), and \(C(3, 5, 7)\) are collinear, then
(Exam: IIT-JEE Year: 2007)
12. The distance between the points \((a, 0, 0)\) and \((0, b, 0)\) is
(Exam: BITSAT Year: 2013)
13. The point \((0, 0, 0)\) is called
(Exam: NEET Year: 2018)
14. The plane \(x = 0\) represents
(Exam: IIT-JEE Year: 1995)
15. The distance of the point \((0, 3, 4)\) from the \(x\)-axis is
(Exam: JEE Main Year: 2020)
16. The coordinates of the point symmetric to \(A(2, -3, 4)\) about the \(xy\)-plane are
(Exam: IIT-JEE Year: 2003)
17. A point lies in the third octant if
(Exam: NEET Year: 2017)
18. The distance of the point \((4, 3, 0)\) from the \(y\)-axis is
(Exam: BITSAT Year: 2011)
19. The coordinates of the centroid of the triangle with vertices \((1, 2, 3)\), \((3, 2, 1)\), and \((2, 4, 6)\) are
(Exam: IIT-JEE Year: 2008)
20. A point \(P\) is equidistant from the planes \(x=0\) and \(y=0\). Its locus is
(Exam: KVPY Year: 2015)
21. The distance between the points \((1, -1, 2)\) and \((-2, 3, -2)\) is
(Exam: JEE Main Year: 2021)
22. The point which lies on both the \(xy\)-plane and the \(yz\)-plane is
(Exam: IIT-JEE Year: 1997)
23. The coordinates of a point dividing the line joining \((2, 2, 2)\) and \((4, 6, 8)\) internally in the ratio \(1:3\) are
(Exam: NEET Year: 2019)
24. The point \((0, -5, 0)\) lies on
(Exam: State Engineering Exam Year: 2016)
25. The locus of a point whose distance from the \(xy\)-plane is constant \(k\) is
(Exam: IIT-JEE Year: 2004)
26. The distance of the point \((1, 2, 2)\) from the \(z\)-axis is
(Exam: BITSAT Year: 2014)
27. The coordinates of the point symmetric to \((3, 4, -5)\) about the origin are
(Exam: IIT-JEE Year: 1999)
28. The point \((x, y, z)\) is at equal distance from the three coordinate planes if
(Exam: Olympiad Year: 2013)
29. The number of octants in three dimensional geometry is
(Exam: NEET Year: 2020)
30. The coordinates of the origin in three dimensional geometry are
(Exam: State CET Year: 2014)
31. The distance between \((0, a, 0)\) and \((0, 0, b)\) is
(Exam: IIT-JEE Year: 2002)
32. A point lies on the \(xz\)-plane if its
(Exam: JEE Main Year: 2018)
33. The distance of the point \((2, 0, 0)\) from the \(yz\)-plane is
(Exam: BITSAT Year: 2010)
34. The midpoint of the line joining \((a, 0, 0)\) and \((0, b, 0)\) is
(Exam: IIT-JEE Year: 1996)
35. The locus of a point moving such that its distance from the origin is constant is
(Exam: Olympiad Year: 2016)
36. The distance of \((0, 4, 3)\) from the \(x\)-axis is
(Exam: NEET Year: 2015)
37. A point lies in the first octant if
(Exam: JEE Main Year: 2017)
38. The coordinates of a point on the \(z\)-axis are
(Exam: State Board Year: 2013)
39. The distance between \((1, 2, 3)\) and the origin is
(Exam: IIT-JEE Year: 2000)
40. The plane \(z=0\) is known as
(Exam: NEET Year: 2014)
41. The point \((2, -3, 0)\) lies in
(Exam: BITSAT Year: 2015)
42. The distance of the point \((0, 0, 7)\) from the origin is
(Exam: JEE Main Year: 2016)
43. The reflection of \((x, y, z)\) in the \(yz\)-plane is
(Exam: IIT-JEE Year: 2006)
44. The locus of a point equidistant from the axes is
(Exam: Olympiad Year: 2012)
45. The coordinates of the foot of the perpendicular from \((3, 4, 5)\) to the \(xy\)-plane are
(Exam: IIT-JEE Year: 1994)
46. The distance of the point \((1, 1, 1)\) from the plane \(x=0\) is
(Exam: NEET Year: 2021)
47. A point whose coordinates satisfy \(x=y=z\) lies on
(Exam: IIT-JEE Year: 2009)
48. The distance between \((a, a, a)\) and the origin is
(Exam: BITSAT Year: 2016)
49. The point \((0, 0, -5)\) lies on
(Exam: State CET Year: 2018)
50. The distance between the points \((1, 0, 0)\) and \((0, 1, 0)\) is
(Exam: IIT-JEE Year: 1993)

Frequently Asked Questions

Three Dimensional Geometry studies the position of points in space using three mutually perpendicular axes.

It involves three independent measurements represented by \(x\), \(y\), and \(z\).

It is a reference framework consisting of three perpendicular axes intersecting at a common point.

The axes are the \(x\)-axis, \(y\)-axis, and \(z\)-axis.

The origin is the point where all three axes intersect and has coordinates \((0,0,0)\).

An ordered triplet \((x,y,z)\) represents the coordinates of a point in three dimensional space.

It represents the perpendicular distance of the point from the \(yz\)-plane.

It represents the perpendicular distance of the point from the \(xz\)-plane.

It represents the perpendicular distance of the point from the \(xy\)-plane.

The three coordinate planes are the \(xy\)-plane, \(yz\)-plane, and \(zx\)-plane.

The equation of the \(xy\)-plane is \(z = 0\).

The equation of the \(yz\)-plane is \(x = 0\).

The equation of the \(zx\)-plane is \(y = 0\).

The distance is \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\).

It is derived using the Pythagorean theorem extended to three dimensions.

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