Chapter 12 Introduction to Three Dimensional Geometry Ex 12.3 Solutions
Question - 1 : - Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6) in the ratio (i) 2:3 internally, (ii) 2:3 externally.
Answer - 1 : -
(i) The coordinates ofpoint R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) internally in the ratio m: n are
.
Let R (x, y, z) be the point that divides the linesegment joining points(–2, 3, 5) and (1, –4, 6) internally in the ratio 2:3
Thus, thecoordinates of the required point are
.
(ii) Thecoordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) externally in the ratio m: n are
.
Let R (x, y, z) be the point that divides the linesegment joining points(–2, 3, 5) and (1, –4, 6) externally in the ratio 2:3
Thus, thecoordinates of the required point are (–8, 17, 3).
Question - 2 : - Given that P (3, 2, –4), Q (5, 4, –6) and R (9, 8, –10) are collinear. Find the ratio in which Q divides PR.
Answer - 2 : -
Let point Q (5, 4, –6)divide the line segment joining points P (3, 2, –4) and R (9, 8, –10) in theratio k:1.
Therefore,by section formula,
Thus,point Q divides PR in the ratio 1:2.
Question - 3 : - Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8).
Answer - 3 : -
Let the YZ plane divide theline segment joining points (–2, 4, 7) and (3, –5, 8) in the ratio k:1.
Hence, bysection formula, the coordinates of point of intersection are given by
On the YZplane, the x-coordinate of any pointis zero.
Thus, theYZ plane divides the line segment formed by joining the given points in theratio 2:3.
Question - 4 : - Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and 
are collinear.
Answer - 4 : -
The given points are A (2,–3, 4), B (–1, 2, 1), and.
Let P bea point that divides AB in the ratio k:1.
Hence, bysection formula, the coordinates of P are given by
Now, wefind the value of k at whichpoint P coincides with point C.
By taking
, we obtain k = 2.
For k = 2, the coordinates of point P are
.
i.e., is a point that divides ABexternally in the ratio 2:1 and is the same as point P.
Hence,points A, B, and C are collinear.
Question - 5 : - Find the coordinates of the points which trisect the line segment joining the points P (4, 2, –6) and Q (10, –16, 6).
Answer - 5 : -
Let A and B be the pointsthat trisect the line segment joining points P (4, 2, –6) and Q (10, –16, 6)
Point Adivides PQ in the ratio 1:2. Therefore, by section formula, the coordinates ofpoint A are given by
Point Bdivides PQ in the ratio 2:1. Therefore, by section formula, the coordinates ofpoint B are given by
Thus, (6,–4, –2) and (8, –10, 2) are the points that trisect the line segment joiningpoints P (4, 2, –6) and Q (10, –16, 6).