NCERT Class 12-Mathematics: Chapter –11 Three Dimensional Geometry Part 10

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Question 24:

Find the equation of the plane through the intersection of the planes and, whose perpendicular distance from origin is unity.

Answer:

and

Question 25:

Show that the points and are equidistant from the plane and lies on opposite side of it.

Answer:

To show that these given points are equidistant from the plane , we first out the mid-point of the points which is .

On substituting by the mid-point in plane, we get

LHS

Hence, the two points lie on opposite sides of the plane are equidistant from the plane.

Question 26:

are two vectors. The position vectors of the points A and C are respectively. Find the position vector of a point P on the line AB and a point Q on the line CD such that is perpendicular to and both.

Answer:

Question 27:

Show that the straight lines whose direction cosines are given by and are at right angles.

Answer:

We have,

And

Eliminating from the both equations, we get

Thus, the direction ratios of two lines are proportional to

Also, the vectors parallel to these lines are respectively,

Question 28:

If are the direction cosines of three mutually perpendicular lines, prove that the line whose direction cosines are proportional to makes equal angles with them.

Answer:

Let

Also, let are the angles between

Similarly,

So, the line whose direction cosines are proportional to make equal to angles with the three mutually perpendicular lines whose direction cosines are respectively.

Objective Type Questions

Choose the Correct Answer from the Given Four Options in Each of the Exercises from 29 to 36

Question 29:

Distance of the point from axis is

(A)

(B)

(C)

(D)

Answer: (D)