NCERT Class 11-Math՚s: Chapter – 12 Introduction to Three Dimensional Geometry Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question 6:

Using distance formula show that the points , and are collinear.


Three points are collinear if the sum of any two distances is equal to the third distance.

Since QR = PQ + PR. Therefore, the given points are collinear.

Question 7:

Find the coordinates of a point equidistant from the four points , , and .


Let be the required point. Then . Now


Hence, the coordinate of the required point are .

Question 8:

Find the point on x-axis which is equidistant from the point and .


The point on the x-axis is of form . Since the points A and B are equidistant from P. Therefore , i.e.. ,

Thus, the point P on the axis is which is equidistant from A and B.

Question 9:

Find the point on y-axis which is at a distance from the point


Let the point P be on y-axis. Therefore, it is of the form . The point is at a distance from . Therefore

Hence, the required point is .

Question 10:

If a parallelepiped is formed by planes drawn through the points and parallel to the coordinate planes, then find the length of edges of a parallelepiped and length of the diagonal.


Length of edges of the parallelopiped are i.e.. , . Length of diagonal is units

Question 11:

Show that the points , and form a right angled isosceles triangle.


Let and be the given three points.



Therefore, is a right angled triangle at . Also . Hence is an isosceles triangle.

Question 12:

Show that the , , and are the vertices of a rhombus.


Let , , and be the four points of a quadrilateral. Here

Note that . Therefore, is a rhombus.

Question 13:

Find the ratio in which the line segment joining the points and is divided by the plane.


Let the joint of and be divided by plane in the ratio at the point . Therefore

Since the point lies on the plane, the y-coordinate should be zero, i.e.. ,

Hence, the required ratio is , i.e.. ; externally in the ratio .

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