NCERT Class 11-Math's: Chapter –12 Introduction to Three Dimensional Geometry Part 4

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Question 14:

Find the coordinate of the point P which is five - sixth of the way from to .

Answer:

Let be the required point, i.e., P, divides AB in the ratio . Then

Question 15:

Describe the vertices and edges of the rectangular parallelopiped with vertex placed in the first octant with one vertex at origin and edges of parallelopiped lie along and axes.

Answer:

The six planes of the parallelopiped are as follows:

Plane OABC lies in the plane. The z-coordinate of every point in this plane is zero. is the equation of this plane. Plane PDEF is parallel to plane and unit distance above it. The equation of the plane is . Plane ABPF represents plane . Plane OCDE lies in the plane and is the equation of this plane. Plane AOEF lies in theplane. The y coordinate of every point in this plane is zero. Therefore, is the equation of plane.

Plane BCDP is parallel to the plane AOEF at a distance .

Edge OA lies on theaxis. The x-axis has equation and .

Edges OC and OE lie on y-axis and z-axis, respectively. The y-axis has its equation , . The z-axis has its equation . The perpendicular distance of the point from the axis is .

The perpendicular distance of the point from y-axis and z-axis are and respectively.

The coordinates of the feet of perpendiculars from the point to the coordinate axes are A, C, E. The coordinates of feet of perpendiculars from the point P on the coordinate planes and are , and . Also, perpendicular

Fig-12.8-The Cooredinates of the feet of perpendicular

The Cooredinates of the Feet of Perpendicular

distance of the point P from the and planes are and , respectively, Fig.12.8.

Question 16:

Let , , be three points forming a triangle. AD, the bisector of ∠ BAC, meets BC in D. Find the coordinates of the point D.

Answer:

Note that

Since AD is the bisector of , We have

i.e., D divides BC in the ratio . Hence, the coordinates of D are

Question 17:

Determine the point in plane which is equidistant from three points and .

Answer:

Since x-coordinate of every point in plane is zero. Let be a point on the plane such that . Now

Simplifying the two equating, we get ,

Here, the coordinate of the point P are .

Objective Type Questions

Choose the Correct Answer Out of Given Four Options in Each of the Examples from 18 to 23 (M.C.Q.)

Question 18:

The length of the foot of perpendicular drawn from the point on axis is

(A)

(B)

(C)

(D)

Answer:

Let l be the foot of perpendicular from point P on the y-axis. Therefore, its x and z-coordinates are zero, i.e., . Therefore, distance between the points and is

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