**NCERT Exemplar Problems Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry**

**Short Answer Type Questions**

**Q1. Locate the following points:**

**(i) (1,-1, 3), **

**(ii) (-1,2,4) **

** (iii) (-2, -4, -7) **

**(iv) (-4,2, -5)**

**Sol:** Given, coordinates

(i) (1,-1, 3),

(ii) (-1,2,4)

(iii) (-2, -4, -7)

(iv) (-4,2, -5)

**Q2. Name the octant in which each of the following points lies.**

**(i) (1,2,3) **

**(ii) (4,-2, 3) **

**(iii) (4,-2,-5) **

**(iv)(4,2,-5)**

**(v) (-4,2,5) **

**(vi) (-3,-1,6) **

**(vii) (2,-4,-7) **

**(viii) (-4, 2,-5)
**

**Sol:** We know that the sign of the coordinates of a point determine the octant in which the point lies. The following table shows the signs of the coordinates in eight octants.

**Q3. Let A, B, C be the feet of perpendiculars from a point P on the x, y,z-axes respectively. Find the coordinates of A, B and C in each of the following where the point P is:**

**(i) (3,4,2) **

**(ii) (-5,3,7) **

**(iii) (4,-3,-5)
**

**Sol:**We know that, on x-axis, y, z = 0, on y-axis, x, z = 0 and on z-axis, x,y = 0. Thus, the feet of perpendiculars from given point P on the axis are as follows.

(i) A(3,0,0),5(0,4,0),C(0,0,2)

(ii) A(-5, 0, 0), B(0, 3, 0), C(0, 0, 7)

(iii) A(4, 0, 0), 5(0, -3, 0), C(0,0, -5)

**Q4. Let A, B, C be the feet of perpendiculars from a point P on the xy, yz and zx- planes respectively. Find the coordinates of A, B, C in each of the following where the point P is**

**(i) (3,4,5) **

**(ii) (-5,3,7) **

**(iii) (4,-3,-5).**

**Sol:** We know that, on xy-plane z = 0, on yz-plane, x = 0 and on zx-plane, y = 0. Thus, the coordinates of feet of perpendicular on the xy, yz and zx-planes from the given point are as follows:

**(i)** A(3,4,0), 5(0,4, 5), C(3,0,5)

**(ii)** A(-5, 3,0), 5(0, 3, 7), C(-5, 0, 7)

**(iii)** A(4, -3, 0), 5(0, -3, -5), C(4,0, -5)

**Q5. How far apart are the points (2,0, 0) and (-3, 0, 0)?**

**Sol:** Given points are A (2, 0, 0) and 5(-3,0, 0).

AB = |2 – (-3)| = 5

**Q6. Find the distance from the origin to (6, 6, 7).**

**Sol:** Distance form origin to the point (6, 6, 7)

**Q8. Show that the point ,4(1, -1, 3), 6(2, -4, 5) and (5, -13, 11) are collinear.**

**Sol:** Given points are ,4(1, -1, 3), 6(2, -4, 5) and C(5, -13, 11).

**Q9. Three consecutive vertices of a parallelogram ABCD are .4(6, -2,4), 6(2,4, -8), C(-2, 2, 4). Find the coordinates of the fourth vertex.
Sol:** Let the coordinates of the fourth vertex D be (x, y, z).

**Q10 .Show that the triangle ABC with vertices .4(0,4,1), 6(2,3, -1) and C(4, 5,0) is right angled.**

**Sol:** The vertices of ∆ABC are A(0,4, 1), 5(2, 3, -1) and C(4, 5, 0).

**Q11. Find the third vertex of triangle whose centroid is origin and two vertices are (2,4,6) and (0, -2, -5).**

**Sol: **Let the third or unknown vertex of ∆ABC be A(x, y, z).

Other vertices of triangle are 5(2,4, 6) and C(0, -2, -5).

The centroid is G(0, 0, 0).

**Q12. Find the centroid of a triangle, the mid-point of whose sides are D (1,2, – 3), E(3,0, l)and F(-l, 1,-4).
Sol:**Given that, mid-points of sides of AABC are D(l, 2, -3), E(3, 0, 1) and F(-l, 1,-4).

**Q14. Three vertices of a Parallelogram ABCD are A(\, 2, 3), B(-A, -2, -1) and C(2, 3, 2). Find the fourth vertex**

Sol: Let the fourth vertex of the parallelogram D(x, y, z).

Mid-point of BD

**Q15. Find the coordinate of the points which trisect the line segment joining the points .A(2, 1, -3) and B(5, -8, 3).
**

**Q16. If the origin is the centroid of a triangle ABC having vertices A(a, 1, 3), B(-2, b, -5) and C(4, 7, c), find the values of a, b, c.**

**Sol:** Vertices of AABC are A(a, 1, 3), B(-2, b, -5) and C(4, 7, c).

Also, the centroid is G(0, 0, 0).

**Q17. Let A(2, 2, -3), 5(5, 6, 9) and C(2, 7, 9) be the vertices of a triangle. The internal bisector of the angle A meets BC at the point Find the coordinates of D.
**

**Long Answer Type Questions**

**Q18. Show that the three points A(2, 3, 4), 5(-l, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which Cdivides**

**Sol: **Given points are A(2, 3, 4), B(-1, 2, -3) and C(-4,1,-10)

**Q19. The mid-point of the sides of a triangle are (1, 5, -1), (0,4, -2) and (2, 3,4). Find its vertices. Also, find the centroid of the triangle.**

** Sol:** Given that mid-points of the sides of AABC are D( 1, 5, -1), E(0, 4, -2) and F(2, 3,4).

**Q20. Prove that the points (0, -1, -7), (2, 1, -9) and (6, 5, -13) are collinear. Find the ratio in which the first point divides the join of the other two.
**

**Sol:**Given points are 4(0, -1, -7), 8(2, 1, -9) and C(6, 5, -13).

**Q21. What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?**

**Sol:** The coordinate of the cube whose edge is 2 units, are:

(2, 0, 0), (2,2, 0), (0, 2, 0), (0, 2,2), (0, 0,2), (2,0, 2), (0, 0, 0) and (2,2, 2)

**Objective Type Questions**

**Q22. The distance of point P(3,4, 5) from the yz-plane is**

**(a) 3 units **

**(b) 4 units **

**(c) 5 units **

**(d) 550**

**Sol:** (a) Given point is P{3,4, 5).

Distance of P from yz-plane = |x coordinate of P| = 3

**Q23. What is the length of foot of perpendicular drawn from the point P(3,4, 5) on y-axis?**

**Q24. Distance of the point (3,4, 5) from the origin (0, 0, 0) is
**

**Q25. If the distance between the points (a,0,1) and (0,1,2) is √27, then the value of a is**

**(a) 5 **

** (b) ± 5 **

** (c) -5 **

**(d) none of these
**

**Q26. x-axis is the intersection of two planes**

**(a) xy and xz **

**(b) yz and zx**

**(c) xy and yz **

**(d) none of these**

**Sol:** (a) We know that, on the xy and xz-planes, the line of intersection is x-axis.

**Q27. Equation of Y-axis is considered as**

**(a) x = 0, y = 0 **

**(b) y = 0, z = 0**

**(c) z = 0, x = 0 **

** (d) none of these**

**Sol:**(c) On the j-axis, x = 0 and z = 0.

**Q28. The point (-2, -3, -4) lies in the**

**(a) First octant **

**(b) Seventh octant**

**(c) Second octant **

**(d) Eighth octant
Sol:** (b) The point (-2, -3, -4) lies in seventh octant.

**Q29. A plane is parallel to yz-plane so it is perpendicular to**

**(a) x-axis **

**(b) y-axis **

**(c) z-axis **

** (d) none of these**

**Sol:** (a) A plane parallel to yz-plane is perpendicular to x-axis.

**Q30. The locus of a point for which y = 0, z = 0 is**

**(a) equation of x-axis **

**(b) equation of y-axis**

**(c) equation at z-axis **

** (d) none of these**

**Sol:** (a) We know that, equation of the x-axis is: y = 0, z = 0 So, the locus of the point is equation of x-axis.

**Q31. The locus of a point for which x = 0 is**

**(a) xy-plane **

**(b) yz-plane**

**(c) zx-plane **

** (d) none of these**

**Sol: **(b) On the yz-plane, x = 0, hence the locus of the point is yz-plane.

**Q32. If a parallelepiped is formed by planes drawn through the points (5,8,10) and (3, 6, 8) parallel to the coordinate planes, then the length of diagonal of the parallelepiped is**

**Q33. L is the foot of the perpendicular drawn from a point P(3, 4, 5) on the xy-plane. The coordinates of point L are**

**(a) (3,0,0) **

**(b) (0,4,5)**

**(c) (3, 0, 5) **

**(d) none of these**

**Sol:** (d) We know that on the xy-plane, z = 0.

Hence, the coordinates of the points L are (3,4, 0).

**Q34. L is the foot of the perpendicular drawn from a point (3, 4, 5) on x-axis. The coordinates of L are**

**(a) (3,0,0) **

**(b) (0,4,0)**

**(c) (0, 0, 5) **

**(d) none of these**

**Sol: **(a) On the x-axis, y = 0 and z = 0.

Hence, the required coordinates are (3, 0,0).

**Fill in the Blanks Type Questions**

**Q35. The three axes OX, OY, OZ determine______ .**

**Sol: **The three axes OX, OY and OZ determine three coordinate planes.

**Q36. The three planes determine a rectangular parallelepiped which has____ of rectangular faces.
**

**Q37. The coordinates of a point are the perpendicular distance from the _____ on the respective axes.**

**Sol:** Given points

**Q38. The three coordinate planes divide the space into _________parts.**

**Sol:** Eight

**Q39. If a point P lies in yz-plane, then the coordinates of a point on yz-plane is of the form_______.**

**Sol:** We know that, on yz-plane, x = 0.So, the coordinates of the required point are (0, y, z).

**Q40. The equation of yz-plane is ______ .**

**Sol:** On yz-plane for any point x-coordinate is zero.

So, yz-plane is locus of point such that x = 0, which is its equation.

**Q41. If the point P lies on z-axis, then coordinates of P are of the form_____.**

**Sol:** On the z-axis, x = 0 and y = 0.

So, the required coordinates are of the form (0, 0, z).

**Q42. The equation of z-axis, are ______.**

**Sol:** Any point on the z-axis is taken as (0, 0, z).

So, for any point on z-axis, we have x = 0 and y = 0, which together represents its equation.

**Q43. A line is parallel to xy-plane if all the points on the line have equal_________.**

**Sol:** A line is parallel to xy-plane if each point P(x, y, z) on it is at same distance from xy-plane.

Distance of point P from xy plane is ‘z’

So, line is parallel to xy-plane if all the points on the line have equal z-coordinate.

**Q44. A line is parallel to x-axis if all the points on the line have equal ______.**

**Sol:** A line is parallel to x-axis if each point on it maintains constant distance from y-axis and z-axis.

So, each point has equal y and z-coordinates. .

**Q45. x = a represents a plane parallel to .**

**Sol:** Locus of point P(x, y, z) is x = a.

Therefore, each point P has constant x-coordinate.

Now, x is distance of point P from yz-plane.

So, here plane x = a is at constant distance ‘a’ from yz-plane and parallel to _yz-plane.

**Q46. The plane parallel to yz-plane is perpendicular to_____ .**

**Sol:** The plane parallel to yz-plane is perpendicular to x-axis.

**Q47. The length of the longest piece of a string that can be stretched straight in a rectangular room whose dimensions are 10, 13 and 8 units are______ .**

**Sol:** Given dimensions are: a = 10, 6=13 andc = 8.

Required length of the string = yja^{2} + b^{2} + c^{2} = ^100 + 169 + 64 = -7333

**Q48. If the distance between the points (a, 2,1) and (1,-1,1) is 5, then a_______ .
**

**Sol:**Given points are (a, 2,1) and (1,-1,1).

**Q49. If the mid-points of the sides of a triangle AB; BC; CA are D(l, 2, – 3), E( 3, 0, 1) and F(-l, 1, -4), then the centroid of the triangle ABC is________ .**

**Sol:** Given that, mid-points of sides of AABC are D( 1, 2, -3), E(3, 0, 1) and F(-l, 1,-4).

**Matching Column Type Questions**

**Q50. Match each item given under the column C _{1} to its correct answer given under column C_{2}.**

Column C, | Column C_{2} |
||

(a) | In xy-plane | (i) | 1st octant |

(b) | Point (2, 3,4) lies in the | (ii) | vz-plane |

(c) | Locus of the points having x coordinate 0 is | (iii) | z-coordinate is zero |

(d) | A line is parallel to x-axis if and only | (iv) | z-axis . |

(e) | If x = 0, y = 0 taken together will represent the | (v) | plane parallel to xy-plane |

(f) | z = c represent the plane | (vi) | if all the points on the line have equal y and z-coordinates. |

(g) | Planes x = a, y = b represent the line | (vii) | from the point on the respective axis. |

00 | Coordinates of a point are the distances from the origin to the feet of perpendiculars | (viii) | parallel to z-axis |

(i) | A ball is the solid region in the space | (ix) | disc |

G) | Region in the plane enclosed by a circle is known as a | 00 | sphere |

**Sol:** (a) In xy-plane, z-coordinate is zero.

(b) The point (2, 3,4) lies in 1st octant.

(c) Locus of the points having x-coordinate zero is yz-plane.

(d) A line is parallel to x-axis if and only if all the points on the line have equal y and z-coordinates.

(e)x = 0, y = 0 represent z-axis

(f) z = c represents the plane parallel to xy-plane.

**(g)** The plane x = a is parallel to yz-plane.

Plane y = b is parallel to xz-plane.

So, planes x = a and y = b is line of intersection of these planes.

Now, line of intersection of yz-plane and xz-plane is z-axis.

So, line of intersection of planes x = a andy = b is line parallel to z-axis.

**(h)** Coordinates of a point are the distances from the origin to the feet of perpendicular from the point on the respective axis.

**(i)** A ball is the solid region in the space enclosed by a sphere.

**(j)** The region in the plane enclosed by a circle is known as a disc.

Hence, the correct matches are:

(a) – (iii), (b) – (i), (c) – (ii), (d) – (vi), (e) – (iv),

(f) – (v), (g) – (viii), (h) – (vii), (i) – (x), (j) – (ix),