Question

If \(P\) divides the line segment joining the points \(A(1,2,-1)\) and \(B(-1,0,1)\) externally in the ratio \(1:2\) and \(Q=(1,3,-1)\), then \(PQ=\)

• A. \(\sqrt{10}\)
• B. \(3\)
• C. \(1\)
• D. \(\sqrt{13}\)

Hint:

For external division of points in 3D, carefully use \[ \left( \frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n}, \frac{mz_2-nz_1}{m-n} \right) \] and then apply the 3D distance formula.

Answer 1

The Correct Option is B

Step 1: State the external section formula.
$P$ divides $AB$ externally in ratio $m:n$: \[P=\left(\frac{mx_2-nx_1}{m-n},\;\frac{my_2-ny_1}{m-n},\;\frac{mz_2-nz_1}{m-n}\right).\]
Step 2: Identify given data.
$A=(1,2,-1)$, $B=(-1,0,1)$, ratio $1:2$ external.
Step 3: Compute $P$.
$P_x=(1\cdot(-1)-2\cdot 1)/(1-2)=(-3)/(-1)=3$. $P_y=(1\cdot 0-2\cdot 2)/(1-2)=(-4)/(-1)=4$. $P_z=(1\cdot 1-2\cdot(-1))/(1-2)=3/(-1)=-3$. So $P=(3,4,-3)$.
Step 4: Compute $PQ$ with $Q=(1,3,-1)$.
\[PQ=\sqrt{(3-1)^2+(4-3)^2+(-3-(-1))^2}=\sqrt{4+1+4}=3.\]
Step 5: Verify each component.
$(3-1)^2=4$, $(4-3)^2=1$, $(-3+1)^2=4$. Sum $=9$, $\sqrt{9}=3$. Correct.
Step 6: State the answer.
\[ \boxed{PQ=3} \]