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Chapter 12 Introduction to Three Dimensional Geometry Ex-12.3 Interview Questions Answers

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 (x1y1z1) and Q (x2y2z2) internally in the ratio mare

.

Let R (x, yz) 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 (x1y1z1) and Q (x2y2z2) externally in the ratio mare.

Let R (x, yz) 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).


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Chapter 12 Introduction to Three Dimensional Geometry Ex-12.3 Contributors

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