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.

Answer:

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 .

Answer:

Let be the required point. Then . Now

Similarly,

Hence, the coordinate of the required point are .

Question 8:

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

Answer:

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

Answer:

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.

Answer:

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.

Answer:

Let and be the given three points.

Here

Now

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.

Answer:

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.

Answer:

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 .