Question

Question 5 A point R with x-coordinate 4 lies on the line segment joining the pointsP (2, –3, 4) and Q (8, 0, 10). Find the coordinates of the point R.

[Hint suppose R divides PQ in the ratio k: 1. The coordinates of the point R are given by

Class X1 - Maths -Introduction to Three Dimensional Geometry Page 279

Answer 1

Let the point R ≡ ( x , y, z ) divides the Line PQ in the ratio K : 1
P------------------------ R ------------ Q
A/C to question ,
x - co-ordinate of R = 4
Hence, R ≡ ( 4, y , z )
Use section formula for x - co-ordinate
x = (mx₂ + nx₁)/(m + n)
4 = ( k × 8 + 1 × 2 )/( k + 1 )
4( k + 1) = (8k + 2)
4k + 4 = 8k + 2
-4k = -2 ⇒ k = 1/2

Hence, the point R divides the Line PQ in the ratio 1 : 2
∴ y - co-ordinate of R = {(1 × 0 + 2 × -3)/(1 + 2)} = (-6)/3 = -2
∴ z - co-ordinate of R = {( 1 × 10 + 2 × 4 )/(1 + 2) = ( 18)/3 = 6

Hence, co-ordinates of point R are ( 4, -2, 6)