Question

In which ratio (3, 4, 5) divides the line segment joining (1, 2, 3) and (4, 5, 6) internally?

(a) 1:2

(b) 2:1

(c) 3:4

(d) 4:3

I got this question in a job interview.

My question is based upon Three Dimensional Geometry topic in section Three Dimensional Geometry of Mathematics – Class 11

Answer 1

Right choice is (b) 2:1

For explanation: The coordinates of a point dividing the line segment joining (x1, y1, z1) and (x2, y2, z2) internally in the ratio m: n is \((\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n},\frac{mz_2+nz_1}{m+n})\).

Let the ratio be k : 1.So, the coordinates of a point dividing the line segment joining (1, 2, 3) and (4, 5, 6) internally in the ratio k: 1 is \((\frac{k*4+1*1}{k+1},\frac{k*5+1*2}{k+1},\frac{k*6+1*3}{k+1})\)

=> \((\frac{k*4+1*1}{k+1},\frac{k*5+1*2}{k+1},\frac{k*6+1*3}{k+1})\) is same as (3, 4, 5).

=> (4k+1)/(k+1) = 3

=> 4k+1 = 3k+3

=> k = 2

So, ratio is 2:1.