Introduction to Three dimensional Geometry – Exercise 12.3 – Class XI

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.

Solution:

(i) The coordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) internally in the ratio m: n are

i.e., x = –⁴/₅ , y = ¹/₅ and z = ²⁷/₅

Thus the coordinates of the required point are (-⁴/₅ , ¹/₅, ²⁷/₅)

(ii) The coordinates of point R that divides the line seqment joining P(x₁­, y₁ , z₁) and Q(x₂, y₂ , z₂) externally in the ratio m:n are

i.e., x = -8 , y =17 and z =3

Thus the coordinates of the required point are (-8, 17, 3)

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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.

Solution:

Let point Q (5, 4, –6) divide the line segment joining points P (3, 2, –4) and R (9, 8, –10) in the ratio k:1. Therefore, by section formula,

9k+3 = 5k + 5

4k = 2

k = ²/₄ = ¹/₂

Thus point divides PR in the ratio 1:2

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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).

Solution:

Let the YZ plane divide the line segment joining points (–2, 4, 7) and (3, –5, 8) in the ratio k:1.

Hence, by section formula, the coordinates of point of intersection are given by

3k – 2 = 0

k = ²/₃

Thus, the YZ plane divides the line segment formed by joining the given points in the ratio 2:3.

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4. Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and C (0, ¹/₃, 2)are collinear.

Solution:

The given points are A (2, –3, 4), B (–1, 2, 1), and C (0, ¹/₃, 2).

Let P be a point that divides AB in the ratio k:1.

Hence, by section formula, the coordinates of P are given by

 

we obtain k = 2.

For k = 2, the coordinates of point P are (0, ¹/₃, 2).

i.e., C (0, ¹/₃, 2) is a point that divides AB externally in the ratio 2:1 and is the same as point P.

Hence, points A, B, and C are collinear.

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5. Find the coordinates of the points which trisect the line segment joining the points P (4, 2, –6) and Q (10, –16, 6).

Solution:

Let A and B be the points that trisect the line segment joining points P (4, 2, –6) and Q(10, –16, 6)

Point A divides PQ in the ratio 1:2. Therefore, by section formula, the coordinates of point A are given by

Thus, (6, –4, –2) and (8, –10, 2) are the points that trisect the line segment joining points P (4, 2, –6) and Q (10, –16, 6).

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