Three Dimensional Geometry - Exemplar Solutions
CBSE Class–12 Mathematics
NCERT Exemplar
Chapter - 11
Three Dimensional Geometry - Short Answer Questions
1. If the direction ratios of a line are 
find the direction cosines of the line.
Sol. The direction cosines are given by
Here 
are respectively.
Therefore, 
i.e., 
i.e. 
are D.C’s of the line.
2 Find the direction cosines of the line passing through the points 
and 
.
Sol. The direction cosines of a line passing through the points 
and 
are
.
Here 
Here D.C.’s are
.
3. If a line makes an angle of 
with the positive direction of x, y, z-axes, respectively, then find its direction cosines.
Sol. The direction cosines of a line which makes an angle of 
with the axes, are 
Therefore, D.C.’s of the line are 
i.e., 
4. The 
of a point on the line joining the points 
is 4. Find its 
.
Sol. Let the point P divide QR in the ratio 
, then the co-ordinate of P are
But of P is 4. Therefore,
Hence, the of P is 
.
5. Find the distance of the point whose position vector is 
from the plane 
Sol. Let 
and d=9
So, the required distance is 
.
6. Find the distance of the point 
from the line 
Sol. Let 
is the given point.
Any point 
on the line is given by 
.
Since 
, we have
Thus 
Hence 
.
7. Find the coordinates of the point where the line through 
crosses the plane passing through three points 
.
Sol. Equation of plane through three points is
i.e., 
or 
Equation of line through is
Any point on line (2) is 
. This point lies on plane (1). Therefore,
i.e., 
Hence the required point is 
.
8. Find the distance of the point (–1, –5, – 10) from the point of intersection of the line 
and the plane 
.
Sol. We have 
and
Solving these two equations, we get 
which gives 
.
Therefore, the point of intersection of line and the plane is 
and the other given point is 
. Hence the distance between these two points is
, i.e. 13
9. A plane meets the co-ordinate axes in 
such that the centroid of the 
is the point 
. Show that the equation of the plane is 
Sol. Let the equation of the plane be
Then the co-ordinate of are 
respectively. Centroid of the is
But co-ordinates of the centroid of the are (given).
Therefore, 
Thus, the equation of plane is
10 Find the angle between the lines whose direction cosines are given by the equations 
and 
.
Sol. Eliminating 
from the given two equations, we get
Now if 
then 
and if 
then 
.
Thus, the direction ratios of two lines are proportional to 
and 
i.e. 
.
So, vectors parallel to these lines are
, respectively.
If 
is the angle between the lines, then
Hence, 
.
11. Find the co-ordinates of the foot of perpendicular drawn from the point 
to the line joining the points 
and 
.
Sol. Let L be the foot of perpendicular drawn from the points to the line passing through B and C as shown in the Fig. 11.2. The equation of line BC by using formula 
, the equation of the line BC is
Comparing both sides, we get
Thus, the co-ordinate of L are 
so, that the direction ratios of the line AL are 
i.e.
Since AL is perpendicular to BC, we have,
The required point is obtained by substituting the value of 
, in (1), which is 
.
12. Find the image of the point 
in the line 
.
Sol. Let 
be the given point and let L be the foot of perpendicular from P to the given line.
The coordinates of a general point on the given line are
If the coordinates of L are 
then the direction ratios of 
are 
.
But the direction ratios of given line which is perpendicular to are 
. Therefore, 
which gives 
. Hence coordinates of 
are 
.
Let 
be the image of in the given line. Then is the mid-point of 
. Therefore, 
Hence, the image of in the given line is 
.
13. Find the image of the point having position vector 
in the plane 
Sol. Let the given point be 
and be the image of 
in the plane 
as shown in the Fig. 11.4.
Then PQ is the normal to the plane. Since PQ passes through P and is normal to the given plane, so the equation of PQ is given by
Since Q lies on the line PQ, the position vector of Q can be expressed as 
Since R is the mid point of PQ, the position vector of R is 
i.e., 
Again, since R lies on the plane 
, we have
Hence, the position vector of Q is 
, i.e., 
.
Choose the correct answer from the given four options in each of the Examples 14 to 19.
14.The coordinates of the foot of the perpendicular drawn from the point 
on the 
are given by
(A) 
(B) 
(C) 
(D) 
Sol. (A) is the correct Answer.
15. is a point on the line segment joining the points 
. If of is 5, then its 
is
(A) 2
(B) 1
(C) –1
(D) –2
Sol. (A) is the correct answer. Let divides the line segment in the ratio of 
of the point may be expressed as 
giving 
so that 
. Thus, of is 
.
16. If α, β, γ are the angles that a line makes with the positive direction of x,yand z axes respectively, then the direction cosines of the line are.
(A) 
(B) 
(C) 
(D) 
Sol. (B) is the correct answer.
17. The distance of a point 
from is
(A) 
(B) 
(C) 
(D) 
Sol. (C) is the correct answer. The required distance is the distance of from 
which is 
.
18. The equations of in space are
(A) 
(B) 
(C) 
(D) 
Sol. (D) is the correct answer. On the and 
are zero.
19. A line makes equal angles with co-ordinate axes. Direction cosines of this line are
(A) 
(B) 
(C) 
(D) 
Sol. (B) is the correct answer. Let the line makes angle 
with each of the axis. Then, its direction cosines are 
.
Since 
. Therefore, 
Fill in the blanks in each of the Examples from 20 to 22.
20. If a line makes angles 
with 
respectively, then its direction cosines are __________.
Sol. The direction cosines are 
21. If a line makes angles with the positive directions of the coordinate axes, then the value of 
is __________.
Sol. Note that
22. If a line makes an angle of 
with each of y and z axes, then the angle which it makes with is __________.
Sol. Let it makes angle with . Then 
which after simplification gives 
.
State whether the following statements are True or False in Examples 23 and 24.
23. The points 
are collinear.
Sol. Let A, B, C be the points respectively. Then, the direction ratios of each of the lines AB and BC are proportional to 
. Therefore, the statement is true.
24. The vector equation of the line passing through the points 
is 
Sol. The position vector of the points are
,
and therefore, the required equation of the line is given by
.
Hence, the statement is true.