Important Questions for CBSE Class 11 Maths Chapter 12 - Introduction to Three Dimensional Geometry

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1 Marks Questions

1. Name the octants in which the following lie. (5,2,3)

Ans. I

2. Name the octants in which the following lie. (-5,4,3)

Ans. II

3. Find the image of (-2,3,4) in the y z plane

Ans. (2, 3, 4)

4. Find the image of (5,2,-7) in the plane

Ans. (5, 2, 7)

5. A point lie on X –axis what are co ordinate of the point

Ans.

6. Write the name of plane in which axis and - axis taken together.

Ans.Plane

7. The point lie in which octants

Ans.

8. The point lie in which plane

Ans.

9. A point is in the XZ plane. What is the value of y co-ordinates?

Ans. Zero

10. What is the coordinates of XY plane

Ans.

11. The point lie in which octants.

Ans. II

12. The distance from origin to point is:

Ans.

4 Marks Questions

1.Given that P(3,2,-4), Q(5,4,-6) and R(9,8,-10) are collinear. Find the ratio in which Q divides PR

Ans. Suppose Q divides PR in the ratio :1. Then coordinator of Q are

But, coordinates of Q are (5,4,-6). Therefore

These three equations give

.

So Q divides PR in the ratio or 1:2

2. Determine the points in plane which is equidistant from these point A (2,0,3) B(0,3,2) and C(0,0,1)

Ans. We know that Z- coordinate of every point on -plane is zero. So, let be a point in -plane such that PA=PB=PC

Now, PA=PB

PA2=PB2

Putting in (i) we obtain

Hence the required points (3,2,0).

3. Find the locus of the point which is equidistant from the point A(0,2,3) and B(2,-2,1)

Ans. Let be any point which is equidistant from A(0,2,3) and B(2,-2,1). Then

PA=PB

PA2=PB2

4. Show that the points A(0,1,2) B(2,-1,3) and C(1,-3,1) are vertices of an isosceles right angled triangle.

Ans. We have

And

Clearly AB=BC and AB2+BC2=AC2

Hence, triangle ABC is an isosceles right angled triangle.

5. Using section formula, prove that the three points A(-2,3,5), B(1,2,3), and C(7,0,-1) are collinear.

Ans.Suppose the given points are collinear and C divides AB in the ratio

Then coordinates of C are

But, coordinates of C are (3,0,-1) from each of there equations, we get

Since each of there equation give the same value of V. therefore, the given points are collinear and C divides AB externally in the ratio 3:2.

6. Show that coordinator of the centroid of triangle with vertices A(), B(), and C() is

Ans. Let D be the mid point of AC. Then

Coordinates of D are

Let G be the centroid of Then G, divides AD in the ratio 2:1. So coordinates of D are

i.e.

7. Prove by distance formula that the points and are collinear.

Ans.Distance

Distance

Distance

The paints A.B.C. are collinear.

8. Find the co ordinate of the point which divides the join of and in the ratio internally externally

Ans.Let paint be the required paint.

(i)For internal division

Required paint

(ii)For external division.

Required point

9. Find the co ordinate of a point equidistant from the four points and

Ans.Let be the required point

According to condition

Now

Similarly

and and are mid points of side respectively,

Then

Adding eq (1),(4) and (7) we get

And

Hence co-ordinate of

10. Find the ratio in which the join the andis divided by the plane Also find the co-ordinate of the point of division

Ans. Suppose plane divides and in the ratio at pain

Then co-ordinate of paint

Point lies on the plane

Points must satisfy the equation of plane

Required ratio 5:7

11. Find the centroid of a triangle, mid points of whose sides are

Ans. Suppose co-ordinate of vertices of are

Adding eq. (3), (6) and (9)

Co-ordinate of centroid

12. The mid points of the sides of a are given by find the co ordinate of A, B and C

Ans. Suppose co-ordinate of point are and respectively let and are mid points of side and respectively

Adding eq. (1), (4) and (7)

Similarly

Subtracting eq. (1), (4) and (7) from (10)

Now subtracting eq. (2), (5) and (8) from (11)

Similarly

co-ordinate of point and are

and

13. Find the co-ordinates of the points which trisects the line segment PQ formed by joining the point and

Ans. Let R and S be the points of trisection of the segment PO. Then

R divides PQ in the ratio 1:2

Co-ordinates of point

=

Similarly

S divider PQ in the ratio 2:1

co-ordinates of point S

14. Show that the point
taken in order form the vertices of a parallelogram. Do these form a rectangle?

Ans.Mid point of PR is

i.e.

also mid point of QS is

i.e.

Then PR and QS have same mid points.

PR and QS bisect each other. It is a Parallelogram.

Now and

diagonals an not equal

are not rectangle.

15. A point R with co-ordinates 4 lies on the line segment joining the points and find the co-ordinates of the point R

Ans. Let the point. R divides the line segment joining the point P and Q in the ratio , Then co-ordinates of Point R

The co-ordinates of point R is 4

co-ordinates of point R

16. If the points are collinear, find the values of P and q

Ans. Given points

are collinear

Let point Q divider PR in the ratio K:1

co-ordinates of point

the value of P and q are 6 and 2.

17. Three consecutive vertices of a parallelogram ABCD are and find forth vertex D

Ans. Given vertices of 11gm ABCD

Suppose co-or dine of forth vertex

Mid point of

Mid point of

Mid point of AC = mid point of BD

Co-ordinates of point

18. If A and B be the points and respectively. Find the eq. of the set points P such that where K is a constant

Ans. Let co-ordinates of point P be

6 Marks Questions

1. Prove that the lines joining the vertices of a tetrahedron to the centroids of the opposite faces are concurrent.

Ans. Let ABCD be tetrahedron such that the coordinates of its vertices are , , and

The coordinates of the centroids of faces ABC, DAB, DBC and DCA respectively

Now, coordinates of point G dividing DG1 in the ratio 3:1 are

Similarly the point dividing CG2, AG3 and BG4 in the ratio 3:1 has the same coordinates.

Hence the point is common to DG1, CG2, AG3 and BG4.

Hence they are concurrent.

2. The mid points of the sides of a triangle are (1,5,-1), (0,4,-2) and (2,3,4). Find its vertices.

Ans. Suppose vertices of ABC are respectively

Given coordinates of mid point of side BC, CA, and AB respectively are D(1,5,-1), E(0,4,-2) and F(2,3,4)

Subtracting eq. from we get

Subtracting eq. from

Similarly

Coordinates of vertices of ABC are A(1,3,-1), B(2,4,6) and C(1,7,-5)

3.  Let and be two points in space find co ordinate of point which divides and in the ratio by geometrically

Ans. Let co-ordinate of Point be which divider line segment joining the point in the ratio

Clearly

Similarly   and

4. Show that the plane divides the line joining the points and in the ratios

Ans. Suppose the plane divides the line joining the points and in the ratio

Plane  Passing through

Hence Proved.

5. Prove that the points are the vertices of a regular tetrahedron.

Ans. To prove O,  A,  B,  C are vertices of regular tetrahedron.

We have to show that

|OA|=|OB|=|OC|=|AB|=|BC|=|CA|

|OA|=unit

|OB|=unit

|OC|=

unit

|AB|=

unit

|BC|=

unit

|CA|=

unit

|AB| = |BC| = |CA| = |OA| = |OB| = |OC| = 2 unit

O, A, B, C are vertices of a regular tetrahedron.

6. If A and B are the points and respectively, then find the locus of P such that 3|PA| = 2|PB|

Ans. Given points and

Supper co-ordinates of point
|PA|=
|PA|=
|PB|=
|PB|=
3|PA| = 2|PB|
9 PA2=4 PB2