Question

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

Answer 1

The given points are P=( 2,−3,4 ) and Q=( 8,0,10 ) .

Let the point R=( 4,y,z ) divide PQ in the ratio k:1 .

Using the section formula,

R( x,y,z )=( m x 2 +n x 1 m+n , m y 2 +n y 1 m+n , m z 2 +n z 1 m+n )

Therefore, using the x coordinate,

4= ( k⋅8+1⋅2 ) k+1 4k+4=8k+2 8k−4k=4−2 4k=2 k= 2 4 = 1 2

Using the value of k in y and z coordinates of R,

y= ( 1 2 ⋅0+1⋅( −3 ) ) 1 2 +1 = 0−3 3 2 = −3⋅2 3 =−2

z=( 1 2 ⋅10+1⋅4 1 2 +1 ) = 5+4 3 2 = 9⋅2 3 =3⋅2 =6

Thus, the required point is R=( 4,−2,6 ) .