Question

If the line joining $A(1,3,4)$ and $B$ is divided by the point $(-2,3,5)$ in the ratio $1:3$, then $B$ is equal to:

• A. $(-11,3,8)$
• B. $(-11,3,-8)$
• C. $(-8,12,20)$
• D. $(13,6,-13)$

Answer 1

The correct option is A

$(-11,3,8)$

Explanation for the correct option

Evaluate the coordinates of the given point based on information provided

Let the unknown coordinates of $B$ be $(x,y,z)$

Other coordinate is given as $A(1,3,4)$ which divides the line joining $A$ and $B$ in the ratio $1:3$ at point $(-2,3,5)$.

According to the section formula, line joining points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ in the ratio $m:n$, divides at the point $(a,b,c)$, such that,

$[a=\frac{(m\times x_1)+(n\times x_2)}{m+n},b=\frac{(m\times y_1)+(n\times y_2)}{m+n},c=\frac{(m\times z_1)+(n\times z_2)}{m+n}]$

Applying section formula, we get,

$$\begin{gathered} -2=\frac{(3\times 1)+(1\times x)}{3+1} \\ \Rightarrow -8=3+x \\ \Rightarrow x=-11 \end{gathered}$$

$$\begin{gathered} 3=\frac{(3\times 3)+(1\times y)}{3+1} \\ \Rightarrow 12=9+y \\ \Rightarrow y=3 \end{gathered}$$

$$\begin{gathered} 5=\frac{(3\times 4)+(1\times z)}{3+1} \\ \Rightarrow 20=12+z \\ \Rightarrow z=8 \end{gathered}$$

So, the coordinates of $B$ are $(-11,3,8)$

Therefore, option (A) is the correct answer.