Question

The ratio in which $\hat{i}+2\hat{j}+3\hat{k}$ divides the join of $-2\hat{i}+3\hat{j}+5\hat{k}$ and $7\hat{i}-\hat{k}$ is

• A. $1:2$
• B. $2:3$
• C. $3:4$
• D. $1:4$

Hint:

To find the ratio of division of a line segment, apply the section formula by equating coordinates.

Answer 1

The Correct Option is B

Step 1: Let the given points be: \[ A(-2,\,3,\,5), \quad B(7,\,0,\,-1) \] and the dividing point be: \[ P(1,\,2,\,3) \]
Step 2: Assume that point $P$ divides $AB$ internally in the ratio $m:n$.
Step 3: By the section formula: \[ P=\left(\frac{m x_2+n x_1}{m+n},\frac{m y_2+n y_1}{m+n},\frac{m z_2+n z_1}{m+n}\right) \]
Step 4: Substitute the values: \[ 1=\frac{7m-2n}{m+n}, \quad 2=\frac{0\cdot m+3n}{m+n}, \quad 3=\frac{-m+5n}{m+n} \]
Step 5: From the second equation: \[ 2(m+n)=3n \Rightarrow 2m= n \]
Step 6: Hence, the ratio is: \[ m:n = 2:3 \]