Question

If the point $(a, 8, -2)$ divides the line segment joining the points $(1, 4, 6)$ and $(5, 2, 10)$ in the ratio $m:n$, then $\dfrac{2m}{n} - \dfrac{a}{3} =$

• A. $-7$
• B. $1$
• C. $-2$
• D. $3$

Hint:

Apply section formula component-wise and carefully simplify ratios.

Answer 1

The Correct Option is B

Use section formula: Let $(x, y, z) = \left(\dfrac{m x_2 + n x_1}{m+n}, \dfrac{m y_2 + n y_1}{m+n}, \dfrac{m z_2 + n z_1}{m+n}\right)$
Given point $(a,8,-2)$ divides $(1,4,6)$ and $(5,2,10)$ in $m:n$
So,
$8 = \dfrac{m\cdot 2 + n \cdot 4}{m+n} = \dfrac{2m + 4n}{m+n}$
$\Rightarrow 8m + 8n = 2m + 4n \Rightarrow 6m = -4n \Rightarrow \dfrac{m}{n} = -\dfrac{2}{3}$
Also, $a = \dfrac{5m + n}{m+n} = \dfrac{5(-2) + 3}{-2 + 3} = \dfrac{-10 + 3}{1} = -7$
Now compute: $\dfrac{2m}{n} - \dfrac{a}{3} = 2 \cdot \left(-\dfrac{2}{3} \right) - \left( \dfrac{-7}{3} \right) = -\dfrac{4}{3} + \dfrac{7}{3} = 1$