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
\[ \frac{2m}{n}-\frac{a}{3}= \]

• A. \(-7\)
• B. \(1\)
• C. \(-2\)
• D. \(3\)

Hint:

In 3D section formula problems, use the coordinate that simplifies the ratio first, then substitute into the remaining coordinates.

Answer 1

The Correct Option is B

Step 1: Use the section formula.
Let the points be
\[ A(1,4,6) \] and
\[ B(5,2,10) \]
The point \((a,8,-2)\) divides the line segment joining \(A\) and \(B\) in the ratio \(m:n\).
Using section formula for the \(y\)-coordinate,
\[ 8=\frac{m(2)+n(4)}{m+n} \]

Step 2: Simplify to find the ratio \(m:n\).
\[ 8(m+n)=2m+4n \]
\[ 8m+8n=2m+4n \]
\[ 6m+4n=0 \]
\[ 3m+2n=0 \]
Thus,
\[ \frac{m}{n}=-\frac23 \]
Hence,
\[ \frac{2m}{n}=2\left(-\frac23\right) \]
\[ =-\frac43 \]

Step 3: Find \(a\) using the \(x\)-coordinate.
Using section formula,
\[ a=\frac{m(5)+n(1)}{m+n} \]
Substitute \(m:n=-2:3\):
\[ a=\frac{(-2)(5)+(3)(1)}{-2+3} \]
\[ =\frac{-10+3}{1} \]
\[ =-7 \]
Thus,
\[ \frac{a}{3}=-\frac73 \]

Step 4: Compute the required expression.
\[ \frac{2m}{n}-\frac{a}{3} \]
\[ =-\frac43-\left(-\frac73\right) \]
\[ =-\frac43+\frac73 \]
\[ =\frac33 \]
\[ =1 \]

Step 5: Final conclusion.
Hence,
\[ \boxed{1} \]