NCERT Class 11-Math's: Chapter –12 Introduction to Three Dimensional Geometry Part 2

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12.2 Solved Examples

Short Answer Type

Question 1:

Locate the points (i)

(ii) in space.

Answer:

(i) To locate the point in space, we move units from along the positive direction of x-axis. Let this point be . From the point A moves units parallel to direction of y-axis. Let this point be . From the point B moves units along positive direction of z-axis. Let this point be Fig.(12.3).

Fig. 12.3-Along positive Direction — Along Positive Direction

(ii) From the origin, move 2 units along the negative direction of x-axis. Let this point be . From the point A move units parallel to negative direction of y-axis.

Let this point be . From B move 3 units parallel to positive direction of z - axis. This is our required point (Fig.12.4.)

Fig.12.4-Parallel Direction — Parallel Direction

Question 2:

Sketch the plane (i) (ii) (iii)

Answer:

(i) The equation of the plan represents the plane and equation of the plane represents the plane parallel to plane at a distance 1 unit above plane. Now, we draw a plane parallel to plane at a distance 1 unit above plane Fig.12.5(a).

(ii) The equation of the plane represents the plane and the equation of the plane represents the plane parallel to plane at a distance 3 unit above plane (Fig. 12.5(b)).

(iii) The equation of the plane represents the plane and z = 3 represents the plane parallel to plane at a distance 3 unit above plane (Fig. 12.5(c)).

Fig. 12.5-draw a plane parallel — Draw a Plane Parallel

Question 3:

Let L, M, N be the feet of the perpendiculars drawn from a point on the and axes respectively. Find the coordinates of L, M and N.

Answer:

Since L is the foot of perpendicular from P on the x-axis, its y and z coordinates are zero. The coordinates of L is . Similarly, the coordinates of M and N are and , respectively.

Question 4:

Let L, M, N be the feet of the perpendicular segments drawn from a point on the and planes, respectively. What are the coordinates of L, M and N?

Answer:

Since L is the foot of perpendicular segment from P on the plane, z-coordinate is zero in the plane. Hence, coordinates of L is. Similarly, we can find the coordinates of and , Fig.12.6.

Fig.12.6-Foot of Perpendicular Segment — Foot of Perpendicular Segment

Question 5:

Let L, M, N are the feet of the perpendiculars drawn from the point on the and planes, respectively. Find the distance of these points L, M, N from the point P, Fig.12.7.

Fig.12.7-Feet of Perpendicular Segment — Feet of Perpendicular Segment

Answer:

L is the foot of perpendicular drawn from the point to the plane. Therefore, the coordinate of the point L is . The distance between the point and is . Similarly, we can find the lengths of the foot of perpendiculars on and plane which are 3 and 4 units, respectively.